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GEMINI — DiagPlot: 2D & 3D ternary phase diagrams
CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 312–316
Contents lists available at ScienceDirect
CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry journal homepage: www.elsevier.com/locate/calphad
GEMINI — DiagPlot: 2D & 3D ternary phase diagrams Bertrand Cheynet ∗ , Catherine Bonnet, Milan Stankov THERMODATA - CNRS, 6 rue du tour de l’eau – Saint Martin d’Hères – 38400, France
article
info
Article history: Received 21 April 2008 Received in revised form 4 September 2008 Accepted 7 September 2008 Available online 1 October 2008 Keywords: Thermodynamics Calphad Phase diagram 3D Chemoinformatics
a b s t r a c t GEMINI is the Gibbs Energy Minimizer of ThermoSuite, a package of chemoinformatics devoted to thermodynamics. In the last years a special effort was made to develop specific procedures to plot 2D & 3D ternary phase diagrams. The Diphasic Domain Approach Method allows us to plot 2D isobarothermic sections in ternary systems with efficiency and robustness. The technique of regular meshing applied to the extremities of the conodes of a set of such isobarothermic ternary sections allows us to go for a trip in the third dimension, giving a deeper understanding of ternary phase diagrams by adding a visual 3D approach. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction The last step in the CALPHAD approach is the graphical representation of phase equilibria. In metallurgy, materials science, chemistry and other fields of physic-chemistry, the ternary phase equilibria diagrams have until now been reduced to isobarothermic sections, more commonly called isotherms, and to massic isobaroplethal sections, more commonly called isopleths. These sections are in fact the cuts by a plane, either horizontal for the isotherms or vertical for the isopleth ones, of a scene 3D built on a basis which is the famous equilateral triangle of the space of composition in standardized barycentric co-ordinates of a ternary system, to which is added the third vertical dimension for the temperature. In this scene are placed objects which are no other than the various domains of mono-, bi- or tri-phasic equilibria as shown on Fig. 1. A remark is essential: a 3D space is a cubic space whereas the space of the 2D ternary diagrams of phase equilibria is barycentric. It will thus be necessary to first transform our space of barycentric co-ordinates into a space of Cartesian co-ordinates. This remark seems anodyne but it is certainly the reason why until now very few 3D ternary diagrams were presented while in the same time the number of 3D softwares do not miss on the market. It is thus obvious to everyone that it is possible to go, under some conditions, from a lot of isopleths to the representation in space (Fig. 2), it is what we carried out. Our work is based on the observation that if someone wants to plot an isobaric temperature-composition section of a ternary
∗
Corresponding author. Tel.: +33 4 76 42 76 90. E-mail address: [email protected] (B. Cheynet). URL: http://diagplot.net (B. Cheynet).
0364-5916/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.calphad.2008.09.007
Fig. 1. Domains in a ternary diagram.
phase equilibria diagram he only needs to plot the diphasic domains. It automatically gives the limits of the monophasic and triphasic fields. In such diagrams each diphasic domain is defined by two lines joining all the extremities of the tie-lines of the domain, also called conodes, and limited by the first and the last one. These tie-lines are segments whose the extremities give the compositions of the two phases in equilibrium in the diphasic domain. They are in the plane of the diagram and can easily be determined by equilibria
B. Cheynet et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 312–316
313
Fig. 4. Follow-up of the curve of the limits of binary domains in a ternary system. Fig. 2. Objects assembled in the scene.
determination of diphasic fields: the Diphasic Domain Approach Method [1]. The procedure developed for the following of diphasic domains in the ternary systems is quite simple. We will use in that case the mediatrice of the conode to progress step-by-step in the domains. It is thus necessary from one of the tops to scan a bordering binary until finding a first conode. Then a step in composition is made in the direction given by the mediatrice of this first conode to determine a next one and so on until leaving the two-phase field. If the next domain is a single-phase field it is necessary to restart on the bordering binary starting from the point where it had been left. If the exit of the diphasic is done on a triphasic domain it is necessary to continue the analysis by taking as first conode each of the two other sides of that triphasic field. The Fig. 4 illustrates this procedure. This method gives excellent results as shown on Fig. 5. Fig. 3. Ternary diagram plotted from the conodes of the diphasic domains.
calculation codes. Then ternary phase diagrams can be plotted only using conodes of the diphasic fields (Fig. 3) and the method used to determine the diphasic domains becomes the main point in ternary phase diagrams plotting. To plot such diagrams, the first idea that comes to mind is to do a fine meshing of the whole surface. But anyone can easily see that it is not the best method because it needs many calculation steps and then it is time consuming, many limits of phases being given in this case by calculations in the vicinity, zone in which equilibria are the most difficult to determine. We proposed a curve following approach adapted to each case which allows a rapid and robust
2. Objects The first stage consists of counting and describing the objects to be integrated into the scene. The first object, which should not be forgotten, does not pose any particular problem, it is the reference frame used in the trade: the basic equilateral triangle and the three vertical axes of the temperature. It will then be necessary to count and identify each phase field mono-, di- or three-phasic, to be able to describe them. This stage is decisive because according to the method used it can be more or less consuming in computing time and more or less robust on the quality of the result. We chose an analysis of the space by successive isothermal cuts. A rather low number of isotherms seems sufficient to cover
Fig. 5. Examples of calculated ternary diagrams.
314
B. Cheynet et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 312–316
Fig. 6. Isothermal sections and the corresponding diagram.
Fig. 7. A, conodes. B, meshing. C, surfacing.
Fig. 8. Diphasic domains.
space, equidistant from 20 to 50 degrees when the various fields present a monotonous evolution, closer when there is appearance and/or disappearance of a domain. We use for the calculation of these isotherms the ‘‘Diphasic Domain Approach Method’’, which proved its velocity and its robustness, with the GEMINI solver [2]. Fig. 6 shows an object, a diphasic domain reconstituted from only 3 calculated isotherms. Obviously the only calculated conodes of the isotherms do not make it possible to reconstitute the diagram with sufficient precision and an intermediate work of infographism is necessary: the meshing. 3. Meshing To obtain a good enough realistic virtual diagram it is necessary to materialize each field by describing its limits by smooth surfaces. Let us recall that only the bi- and three-phase fields are to be treated because the single-phase fields are only the complement of those in the composition/temperature space. The realization of these surfaces is obtained from a grid based on the calculated ends of the conodes (Fig. 7A). But the use of the only calculated conodes is not sufficient and allows to obtain only one
coarse grid, firstly because of irregularity of the number of conodes calculated for each isothermal section, then because of irregularity of the spacing of the conodes on the sections, and finally because of irregularity of the sections in temperature. To obtain a soft and smooth surface the grid needs to be fine and regular (Fig. 7B), so it will be necessary to pass by a specific procedure of constitution of this grid on which a skin will be tended (Fig. 7C). That skin is constituted of the whole of the triangles determined at the time of the meshing, which are coloured and illuminated according to their orientation compared to the ray of incidental light. This procedure will be carried out in two steps: firstly the grid will be made regular on each isotherm, then it will be made regular in temperature. 3.1. Meshing of the diphasic domains The limits of the isothermal section of a two-phase field are thermodynamically described by a continuation of conodes, doublets of points describing segments. The limits of the field are obtained by connecting between them the ends of the conodes. Several forms of fields can be identified (Fig. 8).
B. Cheynet et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 312–316
315
Fig. 9. Diphasic Liquid/U domain in Al–Cu–Mg: conodes, meshing, object.
Fig. 10. Triphasic domain in Ag–B–Ge and section at 1185 K.
Fig. 11. Ternary 3D, Diphasic domain, Isothermal section, Isoplethal section.
Each form has its own characteristics and will have to be treated differently : the 3D ‘‘ring’’ domain has two surfaces, the ‘‘fan’’ three, the ‘‘ribbon’’ four.... Considerations on the first and the last conode of the field, on the starting and arrival points of the conodes make it possible to classify each field by type and to treat it consequently with the adapted procedure. Fig. 9 gives an example of work completed to pass from the conodes resulting from the calculation to the representation of a two-phase field made of several different types: fan, ribbon, moon and sun.
3.2. Meshing of the triphasic domains For everyone a triphasic domain is a simple triangle in an isothermal section, but what happens in a 3D space ? As opposed to what one could think, surfaces of the three-phase fields are not inevitably plane as Fig. 10 which presents a three-phase domain in Ag–B–Ge shows it. On the other hand the isothermal sections of these fields are always reduced to triangles or possibly to segments when the tops are aligned, or only to a point when the tops are confused.
316
B. Cheynet et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 312–316
Fig. 12. Cr–Fe–Mo, Al–Cu–Mg, O–U–Zr.
4. Examples See Figs. 11 and 12.
Acknowledgements This work was supported by ARCELOR Research, CNRS, EDF R&D.
5. Conclusion
References
All these plotting capabilities are included in GEMINI, the Gibbs Energy Minimizer of ThermoSuite. But it is also available as standalone free teachware called ‘‘DiagPlot’’ [3].
[1] B. Cheynet, D. Barbier, S. Ricoud, Journal of Thermal Analysis and Calorimetry 90-2 (2007) 333–336. [2] B. Cheynet, P.Y. Chevalier, E. Fischer, CALPHAD 26-2 (2002) 167–174. [3] http://diagplot.net.
Contents lists available at ScienceDirect
CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry journal homepage: www.elsevier.com/locate/calphad
GEMINI — DiagPlot: 2D & 3D ternary phase diagrams Bertrand Cheynet ∗ , Catherine Bonnet, Milan Stankov THERMODATA - CNRS, 6 rue du tour de l’eau – Saint Martin d’Hères – 38400, France
article
info
Article history: Received 21 April 2008 Received in revised form 4 September 2008 Accepted 7 September 2008 Available online 1 October 2008 Keywords: Thermodynamics Calphad Phase diagram 3D Chemoinformatics
a b s t r a c t GEMINI is the Gibbs Energy Minimizer of ThermoSuite, a package of chemoinformatics devoted to thermodynamics. In the last years a special effort was made to develop specific procedures to plot 2D & 3D ternary phase diagrams. The Diphasic Domain Approach Method allows us to plot 2D isobarothermic sections in ternary systems with efficiency and robustness. The technique of regular meshing applied to the extremities of the conodes of a set of such isobarothermic ternary sections allows us to go for a trip in the third dimension, giving a deeper understanding of ternary phase diagrams by adding a visual 3D approach. © 2008 Elsevier Ltd. All rights reserved.
1. Introduction The last step in the CALPHAD approach is the graphical representation of phase equilibria. In metallurgy, materials science, chemistry and other fields of physic-chemistry, the ternary phase equilibria diagrams have until now been reduced to isobarothermic sections, more commonly called isotherms, and to massic isobaroplethal sections, more commonly called isopleths. These sections are in fact the cuts by a plane, either horizontal for the isotherms or vertical for the isopleth ones, of a scene 3D built on a basis which is the famous equilateral triangle of the space of composition in standardized barycentric co-ordinates of a ternary system, to which is added the third vertical dimension for the temperature. In this scene are placed objects which are no other than the various domains of mono-, bi- or tri-phasic equilibria as shown on Fig. 1. A remark is essential: a 3D space is a cubic space whereas the space of the 2D ternary diagrams of phase equilibria is barycentric. It will thus be necessary to first transform our space of barycentric co-ordinates into a space of Cartesian co-ordinates. This remark seems anodyne but it is certainly the reason why until now very few 3D ternary diagrams were presented while in the same time the number of 3D softwares do not miss on the market. It is thus obvious to everyone that it is possible to go, under some conditions, from a lot of isopleths to the representation in space (Fig. 2), it is what we carried out. Our work is based on the observation that if someone wants to plot an isobaric temperature-composition section of a ternary
∗
Corresponding author. Tel.: +33 4 76 42 76 90. E-mail address: [email protected] (B. Cheynet). URL: http://diagplot.net (B. Cheynet).
0364-5916/$ – see front matter © 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.calphad.2008.09.007
Fig. 1. Domains in a ternary diagram.
phase equilibria diagram he only needs to plot the diphasic domains. It automatically gives the limits of the monophasic and triphasic fields. In such diagrams each diphasic domain is defined by two lines joining all the extremities of the tie-lines of the domain, also called conodes, and limited by the first and the last one. These tie-lines are segments whose the extremities give the compositions of the two phases in equilibrium in the diphasic domain. They are in the plane of the diagram and can easily be determined by equilibria
B. Cheynet et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 312–316
313
Fig. 4. Follow-up of the curve of the limits of binary domains in a ternary system. Fig. 2. Objects assembled in the scene.
determination of diphasic fields: the Diphasic Domain Approach Method [1]. The procedure developed for the following of diphasic domains in the ternary systems is quite simple. We will use in that case the mediatrice of the conode to progress step-by-step in the domains. It is thus necessary from one of the tops to scan a bordering binary until finding a first conode. Then a step in composition is made in the direction given by the mediatrice of this first conode to determine a next one and so on until leaving the two-phase field. If the next domain is a single-phase field it is necessary to restart on the bordering binary starting from the point where it had been left. If the exit of the diphasic is done on a triphasic domain it is necessary to continue the analysis by taking as first conode each of the two other sides of that triphasic field. The Fig. 4 illustrates this procedure. This method gives excellent results as shown on Fig. 5. Fig. 3. Ternary diagram plotted from the conodes of the diphasic domains.
calculation codes. Then ternary phase diagrams can be plotted only using conodes of the diphasic fields (Fig. 3) and the method used to determine the diphasic domains becomes the main point in ternary phase diagrams plotting. To plot such diagrams, the first idea that comes to mind is to do a fine meshing of the whole surface. But anyone can easily see that it is not the best method because it needs many calculation steps and then it is time consuming, many limits of phases being given in this case by calculations in the vicinity, zone in which equilibria are the most difficult to determine. We proposed a curve following approach adapted to each case which allows a rapid and robust
2. Objects The first stage consists of counting and describing the objects to be integrated into the scene. The first object, which should not be forgotten, does not pose any particular problem, it is the reference frame used in the trade: the basic equilateral triangle and the three vertical axes of the temperature. It will then be necessary to count and identify each phase field mono-, di- or three-phasic, to be able to describe them. This stage is decisive because according to the method used it can be more or less consuming in computing time and more or less robust on the quality of the result. We chose an analysis of the space by successive isothermal cuts. A rather low number of isotherms seems sufficient to cover
Fig. 5. Examples of calculated ternary diagrams.
314
B. Cheynet et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 312–316
Fig. 6. Isothermal sections and the corresponding diagram.
Fig. 7. A, conodes. B, meshing. C, surfacing.
Fig. 8. Diphasic domains.
space, equidistant from 20 to 50 degrees when the various fields present a monotonous evolution, closer when there is appearance and/or disappearance of a domain. We use for the calculation of these isotherms the ‘‘Diphasic Domain Approach Method’’, which proved its velocity and its robustness, with the GEMINI solver [2]. Fig. 6 shows an object, a diphasic domain reconstituted from only 3 calculated isotherms. Obviously the only calculated conodes of the isotherms do not make it possible to reconstitute the diagram with sufficient precision and an intermediate work of infographism is necessary: the meshing. 3. Meshing To obtain a good enough realistic virtual diagram it is necessary to materialize each field by describing its limits by smooth surfaces. Let us recall that only the bi- and three-phase fields are to be treated because the single-phase fields are only the complement of those in the composition/temperature space. The realization of these surfaces is obtained from a grid based on the calculated ends of the conodes (Fig. 7A). But the use of the only calculated conodes is not sufficient and allows to obtain only one
coarse grid, firstly because of irregularity of the number of conodes calculated for each isothermal section, then because of irregularity of the spacing of the conodes on the sections, and finally because of irregularity of the sections in temperature. To obtain a soft and smooth surface the grid needs to be fine and regular (Fig. 7B), so it will be necessary to pass by a specific procedure of constitution of this grid on which a skin will be tended (Fig. 7C). That skin is constituted of the whole of the triangles determined at the time of the meshing, which are coloured and illuminated according to their orientation compared to the ray of incidental light. This procedure will be carried out in two steps: firstly the grid will be made regular on each isotherm, then it will be made regular in temperature. 3.1. Meshing of the diphasic domains The limits of the isothermal section of a two-phase field are thermodynamically described by a continuation of conodes, doublets of points describing segments. The limits of the field are obtained by connecting between them the ends of the conodes. Several forms of fields can be identified (Fig. 8).
B. Cheynet et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 312–316
315
Fig. 9. Diphasic Liquid/U domain in Al–Cu–Mg: conodes, meshing, object.
Fig. 10. Triphasic domain in Ag–B–Ge and section at 1185 K.
Fig. 11. Ternary 3D, Diphasic domain, Isothermal section, Isoplethal section.
Each form has its own characteristics and will have to be treated differently : the 3D ‘‘ring’’ domain has two surfaces, the ‘‘fan’’ three, the ‘‘ribbon’’ four.... Considerations on the first and the last conode of the field, on the starting and arrival points of the conodes make it possible to classify each field by type and to treat it consequently with the adapted procedure. Fig. 9 gives an example of work completed to pass from the conodes resulting from the calculation to the representation of a two-phase field made of several different types: fan, ribbon, moon and sun.
3.2. Meshing of the triphasic domains For everyone a triphasic domain is a simple triangle in an isothermal section, but what happens in a 3D space ? As opposed to what one could think, surfaces of the three-phase fields are not inevitably plane as Fig. 10 which presents a three-phase domain in Ag–B–Ge shows it. On the other hand the isothermal sections of these fields are always reduced to triangles or possibly to segments when the tops are aligned, or only to a point when the tops are confused.
316
B. Cheynet et al. / CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry 33 (2009) 312–316
Fig. 12. Cr–Fe–Mo, Al–Cu–Mg, O–U–Zr.
4. Examples See Figs. 11 and 12.
Acknowledgements This work was supported by ARCELOR Research, CNRS, EDF R&D.
5. Conclusion
References
All these plotting capabilities are included in GEMINI, the Gibbs Energy Minimizer of ThermoSuite. But it is also available as standalone free teachware called ‘‘DiagPlot’’ [3].
[1] B. Cheynet, D. Barbier, S. Ricoud, Journal of Thermal Analysis and Calorimetry 90-2 (2007) 333–336. [2] B. Cheynet, P.Y. Chevalier, E. Fischer, CALPHAD 26-2 (2002) 167–174. [3] http://diagplot.net.