PHASE DIAGRAMS

H. P. R. Frederikse

A phase is a structurally homogeneous portion of matter. Regardless of the number of chemical constituents of a gas, there is only one vapor phase. This is true also for the liquid form of a pure substance, although a mixture of several liquid substances may exist as one or several phases, depending on the interactions among the substances. On the other hand a pure solid may exist in several phases at different temperatures and pressures because of differences in crystal structure (Ref. 1). At the phase transition temperature, T_tr, the chemical composition of the solid remains the same, but a change in the physical properties often will take place. Such changes are found in ferroelectric crystals (example BaTiO₃) that develop a spontaneous polarization below T_tr, in superconductors (example Pb) that lose all electrical resistance below the transition point, and in many other classes of solids.

In quite a few cases it is difficult to bring about the phase transition, and the high- (or low-) temperature phase persists in its metastable form. Many liquids remain in the liquid state for shorter or longer periods of time when cooled below the melting point (supercooling). However, often the slightest disturbance will cause solidification. Persistence of the high temperature phase in solid–solid transitions is usually of much longer duration. An example of this behavior is found in white tin; although gray tin is the thermodynamically stable form below T_tr (286.4 K), the metal remains in its undercooled, white tin state all the way to T = 0 K, and crystals of gray tin are very difficult to produce.

A phase diagram is a map that indicates the areas of stability of the various phases as a function of external conditions (temperature and pressure). Pure materials, such as mercury, helium, water, and methyl alcohol are considered one-component systems and they have unary phase diagrams. The equilibrium phases in two-component systems are presented in binary phase diagrams. Because many important materials consist of three, four, and more components, many attempts have been made to deduce their multi­component phase diagrams. However, the vast majority of systems with three or more components are very complex, and no overall maps of the phase relationships have been worked out.

It has been shown during the last 20 to 25 years that very useful partial phase diagrams of complex systems can be obtained by means of thermodynamic modeling (Refs. 2, 3). Especially for complicated, multicomponent alloy systems the CALPHAD method has proved to be a successful approach for producing valuable portions of very intricate phase diagrams (Ref. 4). With this method thermodynamic descriptions of the free energy functions of various phases are obtained that are consistent with existing (binary) phase diagram information and other thermodynamic data. Extrapolation methods are then used to extend the thermodynamic functions into a ternary system. Comparison of the results of this procedure with available experimental data is then used to fine-tune the phase diagram and add ternary interaction functions if necessary. In principle this approximation strategy can be extended to four, five, and more component systems.

The 23 phase diagrams shown below present the reader with examples of some important types of single and multicomponent systems, especially for ceramics and metal alloys. This makes it possible to draw attention to certain features like the kinetic aspects of phase transitions (see Figure 22, which presents a time–temperature–transformation, or TTT, diagram for the precipitation of α-phase particles from the β-phase in a Ti-Mo alloy; Ref. 1, pp. 358–360). The general references listed below and the references to individual figures contain phase diagrams for many additional systems.

General References

1. Ralls, K. M., Courtney, T. H., and Wulff, J., Introduction to Materials Science and Engineering, Chapters 16 and 17, John Wiley & Sons, New York, 1976.
2. Kaufman, L., and Bernstein, H., Computer Calculation of Phase Diagrams, Academic Press, New York, 1970.
3. Kattner, U. R., Boettinger, W. J. B., and Coriell, S. R., Z. Metallkd., 87, 9, 1996. [https://doi.org/10.1515/ijmr-1996-870703]
4. Dinsdale, A. T., Ed., CALPHAD, Vol. 1–20, Pergamon Press, Oxford, 1977–1996 and continuing.
5. Baker, H., Ed., ASM Handbook, Volume 3: Alloy Phase Diagrams, ASM International, Materials Park, OH, 1992.
6. Massalski, T. B., Ed., Binary Alloy Phase Diagrams, Second Edition, ASM International, Materials Park, OH, 1990.
7. Roth. R. S., Ed., Phase Diagrams for Ceramists, Vol. I (1964) to Volume XI (1995), American Ceramic Society, Waterville, OH.

References to Individual Phase Diagrams

• Figure 1. Carbon: Reference 7, Vol. X (1994), Figure 8930. Reprinted with permission.
• Figure 2. Si-Ge : Ref. 5, p. 2.231. Reprinted with permission.
• Figure 3. H₂O (ice): See figure.
• Figure 4. SiO₂: Reference 7, Vol. XI (1995), Figure 9174. Reprinted with permission.
• Figure 5. Fe-O: Darken, L.S., and Gurry, R.W., J. Am. Chem. Soc., 68, 798, 1946. Reprinted with permission. [https://doi.org/10.1021/ja01209a030]
• Figure 6. Ti-O: Reference 5, p. 2.324. Reprinted with permission.
• Figure 7. BaO-TiO₂: Reference 7, Vol. III (1975), Figure 4302. Reprinted with permission.
• Figure 8. MgO-Al₂O₃: Reference 7, Vol. XI (1995), Figure 9239. Reprinted with permission.
• Figure 9. Y₂O₃-ZrO₂: Reference 7, Vol. XI (1995), Figure 9348. Reprinted with permission.
• Figure 10. Si-N-Al-O (Sialon): Reference 7, Vol. X (1994), Figure 8759. Reprinted with permission.
• Figure 11. PbO-ZrO₂-TiO₂ (PZT): Reference 7, Vol. III (1975), Figure 4587. Reprinted with permission.
• Figure 12. Al-Si-Ca-O: Reference 7 (1964), Vol. I, Figure 630. Reprinted with permission.
• Figure 13. Y-Ba-Cu-O: Whitler, J.D., and Roth, R.S., Phase Diagrams for High T_c Superconductors, Figure S-082, American Ceramic Society, Waterville, OH, 1990. Reprinted with permission.
• Figure 14. Al-Cu: Reference 5,p. 2.44. Reprinted with permission.
• Figure 15. Fe-C: Ralls, K.M., Courtney, T.H., and Wulff, J., Introduction to Materials Science and Engineering, Figure 16.13, John Wiley & Sons, New York, 1976. Reprinted with permission.
• Figure 16. Fe-Cr: Reference 5, p. 2.152. Reprinted with permission.
• Figure 17. Cu-Sn: Reference 5, p. 2.178. Reprinted with permission.
• Figure 18. Cu-Ni: Reference 5, p. 2.173. Reprinted with permission.
• Figure 19. Pb-Sn (solder): Reference 5, p. 2.335. Reprinted with permission.
• Figure 20. Cu-Zn (brass): Subramanian, P.R., Chakrabarti, D.J., and Laughlin, D.E., Eds., Phase Diagrams of Binary Copper Alloys, p. 487, ASM International, Materials Park, OH, 1994. Reprinted with permission.
• Figure 21. Co-Sm: Reference 5, p. 2.148. Reprinted with permission.
• Figure 22. Ti-Mo: Reference 5, p. 2.296; Reference 1, p. 359. Reprinted with permission.
• Figure 23. Fe-Cr-Ni: Reference 5, Figure 48. Reprinted with permission.

FIGURE 1. Phase diagram of carbon. (A) Martensitic transition: hex graphite → hex diamond. (B) Fast graphite-to-diamond transition. (C) Fast diamond-to-graphite transition.

FIGURE 2. Si-Ge system.

TABLE 1. Si-Ge System

FIGURE 3. Diagram of the principal phases of ice. Solid lines are measured boundaries between stable phases; dotted lines are extrapolated. Ice IV is a metastable phase that exists in the region of ice V. Ice IX exists in the region below –100 °C and pressures in the range 200–400 MPa. Ice X exists at pressures above 44 GPa. See Table 1 for the coordinates of the triple points, where liquid water is in equilibrium with two adjacent solid phases.

TABLE 2. Crystal Structure, Density, and Transition Temperatures for the Phases of Ice (Cell Parameters in Å)

References

1. Wagner, W., Saul, A., and Pruss, A., J. Phys. Chem. Ref. Data, 23, 515, 1994. [https://doi.org/10.1063/1.555947]
2. Lerner, R.G. and Trigg, G.L., Eds., Encyclopedia of Physics, VCH Publishers, New York, 1990.
3. Donnay, J.D.H. and Ondik, H.M, Crystal Data Determinative Tables, Third Edition, Volume 2, Inorganic Compounds, Joint Committee on Powder Diffraction Standards, Swarthmore, PA, 1973.
4. Hobbs, P.V., Ice Physics, Oxford University Press, Oxford, 1974.
5. Glasser, L., J. Chem. Edu., 81, 414, 2004. [https://doi.org/10.1021/ed081p414]

FIGURE 4. SiO₂ system. Crist = cristobalite; Trid = tridymite.

FIGURE 5. Fe-O system.

TABLE 3. Fe-O System

FIGURE 6. Ti-O system.

TABLE 4. Ti-O System