Analytic Approach to Alloys Thermodynamics: Ternary Cu-Ga-Ni system

Abstract

In this paper are presented the results of the calculation of thermodynamic properties in liquid state for ternary Cu-Ga-Ni alloys using the newest version of general solution model. Calculation was carried out in temperature interval 1473-2073 K, along 3 cross sections from corner of each metal, with ratios between two other metals 1:3, 1:1 and 3:1. Partial and integral molar thermodynamic properties in liquid phase for the Cu-Ga-Ni ternary system are determined, presented and discussed. Calculated data is compared with data available from literature and good agreement between these two sets of data was observed. Additionally, isothermal section of phase diagram at 298 K is calculated using Thermo-Calc software and presence of eleven different phases is detected. Presented thermodynamic data for the Cu-Ga-Ni alloys could be useful for the further assessment of this system and its phase diagram as well as for completing thermodynamic description of these alloys.

Keywords
Ternary Cu-Ga-Ni system; Thermodynamics; Calculation; General solution model

1. Introduction

Thermodynamic properties have always held an important place in the study of metallic systems. In the past, thermodynamic quantities mainly have been studied experimentally, but researchers have sought a way to develop different models that would enable calculation of the thermodynamic properties of studied systems, which would not only significantly lower experimental research expenses, but also speed up the research process. For calculation of thermodynamic properties of multicomponent systems different geometric models have been developed, such as Toop¹, Muggianu², general solution model^3-5 etc.

Among available models, general solution model developed by Chou^3,4 has been proved to be most reasonable one in all aspects. This geometric model has been successfully used by different researchers for calculation of thermodynamic properties^6-10, construction and/or evaluation of phase diagrams^11-13, computing physicochemical properties^14-16 and solving technical problems^17-20. In 2010 Zhang and Chou⁵ published new, improved version of general solution model for calculation of thermodynamic properties of liquid mixtures, based on binary Redlich-Kister type parameters and this version has been used to predict thermodynamic properties of metallic systems in liquid phase²¹.

Ternary Cu-Ga-Ni system has great technical and theoretical significance because:

• copper, gallium and nickel are elements included in developing different high temperature lead free solder materials²²,

• presence of Cu-Ga-Ni alloy is detected during study of Pb-free joints produced by using a diffusion soldering method²³,

• this type of alloys are used to develop new microelectronic interconnect materials²⁴ and

• these alloys are basis for forming different multicomponent shape memory alloys like Ni-Cu-Mn-Ga^25,26 and Ni-Mn-Fe-Cu-Ga²⁷.

Despite all of these facts, there is considerable lack of available data on thermodynamic properties of ternary Cu-Ga-Ni alloys in literature (there is just one recent paper dealing with this topic²⁸), although that type of data is essential in further research of phase diagram and understanding of complex processes which occur during bonding and soldering with gallium based materials. Accordingly, the aim of this work is to conduct thermodynamic analysis of ternary Cu-Ga-Ni system using general solution model.

2. Theoretical fundamentals

Among many available methods for calculation thermodynamic properties of ternary system based on information about constitutive binary systems, Chou's general solution model (GSM)³ has been proved to be most reasonable one in all aspects, overcoming inherent defects of the traditional symmetrical and asymmetrical geometric models. This model breaks down boundaries between symmetrical and asymmetrical systems and generalizes various kinds of situations; also accuracy of calculation has been proven in practical examples^29-32.

Recently, a new, improved version of general solution model based on Redlich-Kister parameters was presented by Zhang and Chou⁵. Since older version of GSM involved a series of integration processes which significantly complicated calculation and considering that a large number of real systems can be approximately fit through a Redlich-Kister polynomial, a new formalism, based on the binary Redlich-Kister type parameters, was presented.

Therefore, this new GSM version is utilized for calculating the thermodynamic properties of Cu-Ga-Ni ternary system.

The basic equation of general solution model for ternary system is:

(1) $\begin{array}{c}\Delta {\mathit{G}}^{\mathit{E}}={\varkappa }_1{\varkappa }_2\sum _{\mathit{i}=0}^{\mathit{n}}{\mathit{L}}_{12}^{\mathit{i}}{\left({\varkappa }_1-{\varkappa }_2+\left(2{\mathit{\xi }}_{12}-1\right){\varkappa }_3\right)}^{\mathit{i}}+\hfill \\ {\varkappa }_2{\varkappa }_3\sum _{\mathit{i}=0}^{\mathit{n}}{\mathit{L}}_{23}^{\mathit{i}}{\left({\varkappa }_2-{\varkappa }_3+\left(2{\mathit{\xi }}_{23}-1\right){\varkappa }_1\right)}^{\mathit{i}}+\hfill \\ {\varkappa }_3{\varkappa }_1\sum _{\mathit{i}=0}^{\mathit{n}}{\mathit{L}}_{31}^{\mathit{i}}{\left({\varkappa }_3-{\varkappa }_1+\left(2{\mathit{\xi }}_{31}-1\right){\varkappa }_2\right)}^{\mathit{i}}\hfill \end{array}$

where L^ν_ij are the Redlich-Kister parameters for the binary system ij, independent from composition and only relying on temperature; ΔG^E is integral molar excess Gibbs energy for ternary system and x_i is mole fraction of the component i.

Similarity coefficient ξ is defined as:

(2) $\begin{array}{c}{\mathit{\xi }}_{12}={\mathit{\eta }}_{\mathit{I}}/\left({\mathit{\eta }}_{\mathit{I}}+{\mathit{\eta }}_{\mathit{II}}\right)\hfill \\ {\mathit{\xi }}_{23}={\mathit{\eta }}_{\mathit{II}}/\left({\mathit{\eta }}_{\mathit{II}}+{\mathit{\eta }}_{\mathit{III}}\right)\hfill \\ {\mathit{\xi }}_{31}={\mathit{\eta }}_{\mathit{III}}/\left({\mathit{\eta }}_{\mathit{I}}+{\mathit{\eta }}_{\mathit{II}}\right)\hfill \end{array}$

And the deviation sum of squares can be calculated using:

(3) $\begin{array}{c}{\mathit{\eta }}_{\mathit{I}}=\sum _{\mathit{i}=0}^{\mathit{n}}\frac{1}{2\left(2\mathit{i}+1\right)\left(2\mathit{i}+3\right)\left(2\mathit{i}+5\right)}{\left({\mathit{L}}_{12}^{\mathit{i}}-{\mathit{L}}_{13}^{\mathit{i}}\right)}^2+\hfill \\ \sum _{\mathit{j}=0}^{\mathit{n}}\sum _{\mathit{k}>\mathit{j}}^{\mathit{n}}\frac{1}{\left(\mathit{j}+\mathit{k}+1\right)\left(\mathit{j}+\mathit{k}+3\right)\left(\mathit{j}+\mathit{k}+5\right)}\left({\mathit{L}}_{12}^{\mathit{j}}-{\mathit{L}}_{13}^{\mathit{j}}\right)\left({\mathit{L}}_{12}^{\mathit{k}}-{\mathit{L}}_{13}^{\mathit{k}}\right)\hfill \\ {\mathit{\eta }}_{\mathit{II}}=\sum _{\mathit{i}=0}^{\mathit{n}}\frac{1}{2\left(2\mathit{i}+1\right)\left(2\mathit{i}+3\right)\left(2\mathit{i}+5\right)}{\left({\mathit{L}}_{21}^{\mathit{i}}-{\mathit{L}}_{23}^{\mathit{i}}\right)}^2+\hfill \\ \sum _{\mathit{j}=0}^{\mathit{n}}\sum _{\mathit{k}>\mathit{j}}^{\mathit{n}}\frac{1}{\left(\mathit{j}+\mathit{k}+1\right)\left(\mathit{j}+\mathit{k}+3\right)\left(\mathit{j}+\mathit{k}+5\right)}\left({\mathit{L}}_{21}^{\mathit{j}}-{\mathit{L}}_{23}^{\mathit{j}}\right)\left({\mathit{L}}_{21}^{\mathit{k}}-{\mathit{L}}_{23}^{\mathit{k}}\right)\hfill \\ {\mathit{\eta }}_{\mathit{III}}=\sum _{\mathit{i}=0}^{\mathit{n}}\frac{1}{2\left(2\mathit{i}+1\right)\left(2\mathit{i}+3\right)\left(2\mathit{i}+5\right)}{\left({\mathit{L}}_{31}^{\mathit{i}}-{\mathit{L}}_{32}^{\mathit{i}}\right)}^2+\hfill \\ \sum _{\mathit{j}=0}^{\mathit{n}}\sum _{\mathit{k}>\mathit{j}}^{\mathit{n}}\frac{1}{\left(\mathit{j}+\mathit{k}+1\right)\left(\mathit{j}+\mathit{k}+3\right)\left(\mathit{j}+\mathit{k}+5\right)}\left({\mathit{L}}_{31}^{\mathit{j}}-{\mathit{L}}_{32}^{\mathit{j}}\right)\left({\mathit{L}}_{31}^{\mathit{k}}-{\mathit{L}}_{32}^{\mathit{k}}\right)\hfill \end{array}$

where for all L_ij parameters is valid relation L^k_ij = (-1)^kL^k_ji.

In all equations as given, L^x_ij (x=i, j or k) are the Redlich-Kister parameters for the binary system ij, independent from composition and only relying on temperature; ΔG^(E) is integral molar excess Gibbs energy for ternary system and x_i is mole fraction of the component i. Number n is equal to maximum number of components in system (in this case n=3) and coefficients i, j and k are always numbers between zero and n.

Partial thermodynamic quantities are calculated according to the equations:

(4) ${\mathit{G}}_{\mathit{i}}^{\mathit{E}}={\mathit{G}}^{\mathit{E}}+\left(1-{\varkappa }_{\mathit{i}}\right)\left(\partial {\mathit{G}}^{\mathit{E}}/\partial {\varkappa }_{\mathit{i}}\right)=\mathit{RT}\ln {\mathit{\gamma }}_{\mathit{i}}$

and

(5) ${\mathit{a}}_{\mathit{i}}={\varkappa }_{\mathit{i}}{\mathit{\gamma }}_{\mathit{i}}$

In above mentioned equations G^E_i is partial molar excess Gibbs energy of component i, a_i is activity of component i, γ_i is activity coefficient of component i, T is temperature, R is gas constant (value 8.314 JK^-1mol^-1), G^E is Gibbs energy for whole system, dependent on composition.

3. Results and discussion

Basic thermodynamic data on the constituent binary subsystems Cu-Ga, Ga-Ni and Cu-Ni, needed for calculation of thermodynamic properties in the investigated Cu-Ga-Ni system, were taken from available literature data^33-35, and presented in the form of Redlich-Kister parameters in Table 1.

Table 1
Redlich-Kister parameters for constitutive binary systems (in J/mol)

System ij	Loij (T)	L1ij (T)	L2ij (T)
Cu-Ga33	-58110.5+154.5439*T-18.3753*T*lnT	-33884.7+1.9151*T	-11256.9
Ga-Ni34	-122488.59+35.72*T	-29685+14*T	-30751.9+22.1*
Cu-Ni35	12048.61+1.29893*T	-1861.61+0.94201*T	0

Thermodynamic properties of ternary Cu-Ga-Ni system has been investigated in 9 sections (Figure 1), taken from Cu, Ga and Ni corner, respectively, with ratios 1:3 and 3:1, and with molar content of 0-0.9 for the third component.

Figure 1
Schematic diagram of the investigated concentration regions in ternary Cu-Ga-Ni system.

The calculated integral molar Gibbs excess energies (ΔG^E) and activities of the investigated system Cu-Ga-Ni, along selected sections and at given temperatures, are presented in Figures 2 - 4. All thermodynamic properties calculated in this work are related to the liquid phase.

Figure 2
Results of thermodynamic calculation according to GSM in temperature range of 1473-2073K for cross-sections from copper corner: a) integral molar excess Gibbs energy (ΔG^E) and b) copper activity for cross section Ga:Ni=1:3; c) ΔG^E and d) copper activity for cross section Ga:Ni=1:1; e) ΔG^E and f) copper activity for cross section Ga:Ni=3:1.

Figure 3
Results of thermodynamic calculation according to GSM in temperature range of 1473-2073K for cross-sections from gallium corner: a) integral molar excess Gibbs energy (ΔG^E) and b) gallium activity for cross section Cu:Ni=1:3; c) ΔG^E and d) gallium activity for cross section Cu:Ni=1:1; e) ΔG^E and f) gallium activity for cross section Cu:Ni=3:1.

Figure 4
Results of thermodynamic calculation according to GSM in temperature range of 1473-2073K for cross-sections from nickel corner: a) integral molar excess Gibbs energy (ΔG^E) and b) nickel activity for cross section Cu:Ga=1:3; c) ΔG^E and d) nickel activity for cross section Cu:Ga=1:1; e) ΔG^E and f) nickel activity for cross section Cu:Ga=3:1.

Calculation of thermodynamic properties for ternary system Cu-Ga-Ni was performed using the general solution model. Excess integral Gibbs energy values for investigated sections from nickel corner are negative, with minimum values up to -14 kJ/mol, for gallium corner those values are between 4 kJ/mol and -14 kJ/mol, while for the investigated section from the corner of copper Gibbs energy is within 0 kJ/mol to -18 kJ/mol.

For cross section Ga:Ni=1:3 copper activity shows positive deviation from Rault's law which with increase of copper content in alloy slowly decreases and for x_Cu > 0.6 coincides with the ideal state line. Other investigated cross sections from copper corner shows variable character of deviation from Rault's law, in section Ga:Ni=1:1 copper activity has positive deviation up to x_Cu = 0.4, and for Ga:Ni=3:1 border copper content at which deviation changing from positive to negative occurs is x_Cu = 0.2.

For gallium activity is characteristic pronounced negative deviation from Rault's law for all investigated sections, but for alloys with high gallium content (x_Ga > 0.9) deviation coincides with the line of ideal state and even becomes slightly positive.

Nickel activity shows a negative deviation from the Rault's law for Cu:Ga=1:3 and Cu:Ga=1:1 sections, up to x_Ni > 0.8, while for section Cu:Ga=3:1 nickel activities positively derivate from ideal state line.

Because of notable absence of experimental data related to thermodynamics of this system in literature, data obtained by calculation are compared with data obtained using Redlich-Kister-Muggianu (RKM) model²⁸ (Figure 5). Compared values show very good agreement.

Figure 5
Dependence of copper (a), gallium (b) and nickel (c) activities from composition at 1773 K, predicted according to GSM, compared with literature data²⁸.

Additionally, isothermal section of ternary Cu-Ga-Ni phase diagram at 298K (Figure 6) is calculated using Thermo-Calc software^36,37 developed by Thermo-Calc Software AB, based on CALPHAD^38,39 approach to calculation of phase diagrams. Thermo-Calc is a powerful and flexible software package based upon a powerful Gibbs Energy Minimizer and developed for performing various kinds of thermodynamic and phase diagram calculations. Calculations are based on thermodynamic data which is supplied in a database SSOL5. Wide selections of high-quality databases for various purposes that include many different materials are available and data included in databases is provided by experts through critical assessment and systematic evaluation of experimental and theoretical data, following the well-established CALPHAD technique.

Eleven different phases can be identified in isothermal section of ternary Cu-Ga-Ni system (Figure 6) and five of them are intermetallic compounds present in phase diagram of binary Ga-Ni binary system⁴⁰. Phase 1 is consisted from nickel with some gallium and copper (due to mutual solubility), phase 2 from copper with little amount of nickel, and phase 3 is pure gallium. It is not surprising to detect liquid phase at room temperature, considering that melting temperature of gallium is just 29.78 ºC (302.93 K)⁴¹.

Figure 6
Calculated isothermal section of ternary Cu-Ga-Ni phase diagram at 298K (obtained using Thermocalc 3.0).

4. Conclusion

Calculation of thermodynamic properties of the Cu-Ga-Ni system has been conducted using general solution model, in temperature interval from 1473 to 2073 K.

Based on this, excess molar Gibbs energies and activity of all components were calculated. Calculated excess integral Gibbs energy for investigated sections is mostly negative, with values in range from 4 kJ/mol to -18 kJ/mol. Activity of nickel and gallium shows negative deviation from Rault's law for all investigated sections, but for alloys with high content of nickel or gallium, deviation becomes slightly positive. Deviation of copper activity values from Rault's law depends from share of two other metals, and for alloy which contain same amount of nickel and gallium (Ga:Ni=1:1) copper activity negatively deviates from ideal conditions after x_Cu = 0.4.

Isothermal section of ternary Cu-Ga-Ni phase diagram at 298K is calculated using Thermo-Calc software, based on thermodynamic data which is supplied in a database SSOL5 and presence of eleven different phases is detected.

Presented thermodynamic data for the Cu-Ga-Ni alloys could be useful for the further assessment of this system and its phase diagram as well as for completing thermodynamic description of these alloys.

5. Acknowledgement

The authors are grateful to the Ministry of Education, Science and Technological Development of the Republic of Serbia, Projects 34005: "Development of ecological knowledge-based advanced materials and technologies for multifunctional application" and 172037 "Modern multi-component metal systems and nanostructured materials with different functional properties", for financial support.

6. References

• ¹
  Toop GW. Predicting ternary activities using binary data. Transactions of Metallurgical Society of AIME 1965;233:850-854.
• ²
  Muggianu YM, Gambino M, Bross JP. Enthalpies of formation of liquid alloys bismuth-gallium-tin at 723k. Choice of an analytical representation of integral and partial thermodynamic functions of mixing for this ternary-system. Journal de Chimie Physique et de Physico-Chimie Biologique. 1975;72(1):83-88.
• ³
  Chou KC. A general solution model for predicting ternary thermodynamic properties. Calphad 1995;19(3):315-325.
• ⁴
  Chou KC, Wei SK. A new generation solution model for predicting thermodynamic properties of a multicomponent system from binaries. Metallurgical and Materials Transactions B 1997;28(3):439-445.
• ⁵
  Zhang GH, Chou KC. General formalism for new generation geometrical model: application to the thermodynamics of liquid mixtures. Journal of Solution Chemistry 2010;39(8):1200-1212.
• ⁶
  Trumić B, Zivković D, ZivkovićŽ, Manasijević D. Comparative thermodynamic analysis of the Pb-Au_0.7Sn_0.3 section in the Pb-Au-Sn ternary system. Thermochimica Acta 2005;435(1):113-117.
• ⁷
  Zivković D, ZivkovićŽ, Tasić I. Comparative thermodynamic study of the Pb-Bi₂ Mg₃ system. Thermochimica Acta 2000;362(1-2):113-120.
• ⁸
  Manasijević D, Zivković D, ZivkovićŽ,. Prediction of the thermodynamic properties for the Ga-Sb-Pb ternary system. Calphad 2003;27(4):361-366.
• ⁹
  Minić D, Zivković D, ZivkovićŽ. Thermodynamic and structural analysis of the Pb-InSb system. Thermochimica Acta 2003;400(1-2):143-152.
• ¹⁰
  Zivković D, ZivkovićŽ, Vučinić B. Comparative thermodynamic analysis of the Bi-Ga_0.1Sb_0.9 section in the Bi-Ga-Sb system. Journal of Thermal Analysis and Calorimetry 2000;61(1):263-271.
• ¹¹
  Wan JF, Chen SP, Hsu TY. The stability of transition phases in Fe-Mn-Si based alloys. Calphad 2001;25(3):355-362.
• ¹²
  Jin XJ, Dunne D, Allen SM, O'Handley RC, Hsu TY. Thermodynamic consideration of the effect of alloying elements on martensitic transformation in Fe-Mn-Si based alloys. Journal de Physique IV France 2003;112:369-372.
• ¹³
  Liu YJ, Liang D. A contribution to the Al-Pb-Zn ternary system. Journal of Alloys and Compounds 2005;403(1-2):110-117.
• ¹⁴
  Wang LJ, Chou KC, Seetharaman S. A comparison of traditional geometrical models and mass triangle model in calculating the surface tensions of ternary sulphide melts. Calphad 2008;32(1):49-55.
• ¹⁵
  Prasad LC, Mikula A. Surface segregation and surface tension in Al-Sn-Zn liquid alloys. Physica B: Condensed Matter 2006;373(1):142-149.
• ¹⁶
  Yan LJ, Zheng SB, Ding GJ, Xu GT, Qiao ZY. Surface tension calculation of the Sn-Ga-In ternary alloy. Calphad 2007;31(1):112-119.
• ¹⁷
  Sun ZB, Guo J, Li Y, Zhu YM, Li Q, Song XP. Effects of Ti addition on the liquid-phase separation of Cu₇₁ Cr₂₉ alloy during rapid cooling. Metallurgical and Materials Transactions A 2008;39(5):1054-1059.
• ¹⁸
  Prasad LC, Mikula A. Thermodynamics of liquid Al-Sn-Zn alloys and concerned binaries in the light of soldering characteristics. Physica B: Condensed Matter 2006;373(1):64-71.
• ¹⁹
  Gao F, Takemoto T, Nishikawa H, Komatsu A. Microstructure and mechanical properties evolution of intermetallics between Cu and Sn-_3.5Ag solder doped by Ni-Co additives. Journal of Electronic Materials 2006;35(5):905-911.
• ²⁰
  Ma X, Yoshida F. Interaction relation in 60Sn-Pb-0.05La ternary solder alloy. Materials Letters 2002;56(4):441-445.
• ²¹
  Živković D, Du Y, Talijan N, Kostov A, Balanović L. Calculation of thermodynamic properties in liquid phase for ternary Al-Ni-Zn alloys. Transactions of Nonferrous Metals Society of China 2012;22(12):3059-3065.
• ²²
  Kroupa A, Dinsdale A, Watson A, Vreštal JJ, Zemanova A, Broz P. The thermodynamic database COST MP0602 for materials for high-temperature lead-free soldering. Journal of Mining and Metallurgy. Section B: Metallurgy 2012;48(3):339-346.
• ²³
  Sommadossi S, Troiani HE, Fernández-Guillermet A. Diffusion soldering using a Gallium metallic paste as solder alloy: study of the phase formation systematics. Journal of Materials Science 2007;42:9707-9712.
• ²⁴
  Baldwin DF, Deshmukh RD, Hau CS. Gallium alloy interconnects for flip-chip assembly applications. IEEE Transactions on Components and Packaging Technologies 2000;23(2):360-366.
• ²⁵
  Wang J, Li P, Jiang C. Phase stability and magnetic properties of Ni_50-x Cu_x Mn₃₁ Ga₁₉ alloys. Intermetallics 2013;34:14-17.
• ²⁶
  Li Y, Wang J, Jiang C. Study of Ni-Mn-Ga-Cu as single-phase wide-hysteresis shape memory alloys. Materials Science and Engineering: A 2011;528(22-23):6907-6911.
• ²⁷
  Khan M, Gautam B, Pathak A, Dubenko I, Stadler S, Ali S. Intermartensitic transitions in Ni-Mn-Fe-Cu-Ga Heusler alloys. Journal of Physics: Condensed Matter 2008;20(50):505206.
• ²⁸
  Gomidželović L, Požega E, Kostov A, Krstić V. Thermodynamic analisys of Cu-Ga-Ni system using RKM model. Copper 2014;39(1):1-8. (in Serbian).
• ²⁹
  Balanović L, Živković D, Mitovski A, Manasijević D, ŽivkovićŽ. Calorimetric investigations and thermodynamic calculation of Zn-Al-Ga system Journal of Thermal Analysis and Calorimetry. 2011;103(3):1055-1061.
• ³⁰
  Gomidželović L, Mihajlović I, Kostov A, Živković D. Cu-Al-Zn System: Calculation of thermodynamic properties in liquid phase. Hemijska industrija. 2013;67(1):157-164.
• ³¹
  Živković D, Minić D, Manasijević D, Kostov A, Talijan N, Balanović L, et al. Thermodynamic analysis and characterization of alloys in Bi-Cu-Sb system. Journal of Mining and Metallurgy. Section B: Metallurgy. 2010;46(1):105-111.
• ³²
  Živković D, Holjevac Grgurić T, Gojić M, Čubela D, Stanojević Šimišić Z, Kostov A , et al. Calculation of Thermodynamic Properties of Cu-Al-(Ag,Au) Shape Memory Alloy Systems. Transactions of the Indian Institute of Metals. 2014;67(2):285-289.
• ³³
  Li JB, Ji LN, Liang JK, Zhang Y, Luo J, Li CR, et al. A thermodynamic assessment of the copper-gallium system. Calphad 2008;32(2):447-453.
• ³⁴
  Yuan WY, Qiao ZY, Ipser H, Eriksson G. Thermodynamic assessment of the Ni-Ga system. Journal of Phase Equilibria and Diffusion 2004;25(1):68-74.
• ³⁵
  an May S. Thermodynamic re-evaluation of the Cu-Ni system. Calphad 1992;16(3):255-260.
• ³⁶
  Andersson JO, Helander T, Höglund L, Shi P, Sundman B. Thermo-Calc & DICTRA, computational tools for materials science. Calphad 2002;26(2):273-312.
• ³⁷
  Thermo-Calc Software. Available from: <http://www.thermocalc.com>. Access in: 27/7/2016.
  » http://www.thermocalc.com
• ³⁸
  Lukas H, Fries SG, Sundman B. Computational Thermodynamics: The Calphad Method New York: Cambridge University Press; 2007.
• ³⁹
  CALPHAD. Available from: <http://www.calphad.org>. Access in: 27/7/2016.
  » http://www.calphad.org
• ⁴⁰
  FactSage 7.0. Available from: <http://www.factsage.cn>. Access in: 27/7/2016.
  » http://www.factsage.cn
• ⁴¹
  Galium. Chemichal Elements.com. Available from: <http://www.chemicalelements.com/elements/ga.html>. Access in: 27/7/2016.
  » http://www.chemicalelements.com/elements/ga.html

Data availability

Data citations

FactSage 7.0. Available from: <http://www.factsage.cn>. Access in: 27/7/2016.

Publication Dates

• Publication in this collection
  18 Aug 2016
• Date of issue
  Sep-Oct 2016

History

• Received
  23 Apr 2015
• Reviewed
  02 Dec 2015
• Accepted
  13 July 2016