Unary-Phase Diagrams

In the case of a unary or one-component system, only temperature and pressure may be varied, so the coordinates of unary phase diagrams are pressure and temperature. In a typical unary diagram, as shown in Figure 3.11, the temperature is chosen as the horizontal axis by convention, although in binary diagrams temperature is chosen as the vertical axis. However, for a one-component system, the phase rule becomes F=l-P+2 = 3-P. This means that the maximum number of phases in equilibrium is three when F equals zero. This is illustrated in Figure 3.11 which has three areas, i.e., solid, liquid, and vapour In any... 

Figure 3.11 Phase diagram for a one-component system (unary phase diagram).

A phase diagram is often considered as something which can only be measured directly. For example, if the solubility limit of a phase needs to be known, some physical method such as microscopy would be used to observe the formation of the second phase. However, it can also be argued that if the thermodynamic properties of a system could be properly measured this would also define the solubility limit of the phase. The previous sections have discussed in detail unary, single-phase systems and the quantities which are inherent in that sjrstem, such as enthalpy, activity, entropy, etc. This section will deal with what happens when there are various equilibria between different phases and includes a preliminary description of phase-diagram calculations. 

The unary phase diagram is seldom used in solid state syntheses. However, the unary diagram forms the basis for the phase diagrams of multicomponent systems. Since there are no composition variables, the only externally controllable variables in a unary system are simply the temperature and pressure. For this... 

Univariant equilibrium for which there is one degree of freedom, represents the equilibrium between two co-existing phases. Since there is only one degree of freedom, choosing a value for one external variable, e.g. temperature, determines the remaining variable in a dependent manner, and the locus of points represented on the phase diagram for univariant behavior must lie on a line or curve. Thus the curves on the unary phase diagram represent solid-liquid, solid-vapor, solid-solid, and liquid-vapor equilibrium. 

Because there is an added term, the composition, binary systems are inherently more complex than unary systems. In order to completely represent the phase diagram of a binary system a three dimensional pressure-temperature-composition (P-T-x) diagram can be constructed. However, it is a more common... 

Identify the number of components present, the number of phases present, the composition of each phase, and the quantity of each phase from unary, binary, and ternary phase diagrams—that is, apply the Gibbs Phase Rule. 

Figure 2.1 Unary phase diagram (top) and Gibbs free energy plot (bottom) for elemental sulfur. Reprinted, by permission, from D. R. Gaskell, Introduction to Metallurgical Thermodynamics, 2nd ed., p. 178, Copyright 1981 by Hemisphere Publishing Corporation.

Figure 2.2 Temperature-Pressure unary phase diagram for carbon. From K. M. Ralls, T. H. Courtney, and J. Wulff, Introduction to Materials Science and Engineering. Copyright 1976 by John Wiley Sons, Inc. This material is used by permission of John Wiley Sons, Inc.

Much of what we need to know abont the thermodynamics of composites has been described in the previous sections. For example, if the composite matrix is composed of a metal, ceramic, or polymer, its phase stability behavior will be dictated by the free energy considerations of the preceding sections. Unary, binary, ternary, and even higher-order phase diagrams can be employed as appropriate to describe the phase behavior of both the reinforcement or matrix component of the composite system. At this level of discussion on composites, there is really only one topic that needs some further elaboration a thermodynamic description of the interphase. As we did back in Chapter 1, we will reserve the term interphase for a phase consisting of three-dimensional structure (e.g., with a characteristic thickness) and will use the term interface for a two-dimensional surface. Once this topic has been addressed, we will briefly describe how composite phase diagrams differ from those of the metal, ceramic, and polymer constituents that we have studied so far. 

Unlike the unary, binary, and ternary phase diagrams of the previous sections, there are no standardized guidelines for presenting phase information in composite systems. This... 

Unary phase diagrams are two-dimensional graphs that display the phases of singlecomponent systems (e.g. elements) as a function of both temperature (abscissa) and pressure (ordinate). Since there is only one component, it is not necessary to specify composition. Figure 11.2 shows the phase diagram for sulfur, which exists in two allotropes at 1 atm of pressure, rhombic (T < 368 K) and monoclinic T > 368 K). 

It is possible in many cases to predict highly accurate phase equihbria in multi-component systems by extrapolation. Experience has shown extrapolation of assessed (n — 1) data into an nth order system works well for n < 4, at least with metallurgical systems. Thus, the assessment of unary and binary systems is especially critical in the CALPHAD method. A thermodynamic assessment involves the optimization of aU the parameters in the thermodynamic description of a system, so that it reproduces the most accurate experimental phase diagram available. Even with experimental determinations of phase diagrams, one has to sample compositions at sufficiently small intervals to ensure accurate reflection of the phase boundaries. 

Through their parallel and independent efforts on both sides of the Atlantic, which began in the 1950s with mathematically modeling known phase diagrams for unary and binary systems, Kaufman and Hillert are considered founding fathers of the CALPHAD method, the field of computational thermodynamics concerned with the extrapolation of phase diagrams for multicomponent systems. (Source L. P. Kaufman, personal communication, February 08, 2004.)... 

The subject matter is introduced by a short exposition of the Gibbs phase rule in Sec. 8.2. Unary component systems are discussed in Sec. 8.3. Binary and ternary systems are addressed in Secs. 8.4 and 8.5, respectively. Sec. 8.6 makes the connection between free energy, temperature, and composition, on one hand, and phase diagrams, on the other. 

The objective of this section is much less ambitious and can be formulated as follows If the free-energy function for all phases in a given system were known as a function of temperature and composition, how could one construct the corresponding phase diagram In other words, what is the relationship between free energies and phase diagrams Two examples are considered below polymorphic transformation in unary systems and complete solid solubility. 

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. 

In a one-component, or unary, system, only one chemical component is required to describe the phase relationships, for example, iron (Fe), water (H2O) or methane (CH4). There are many one-component systems, including all of the pure elements and compounds. The phases that can exist in a one-component system are limited to vapour, liquid and solid. Phase diagrams for one-component systems are specified in terms of two variables, temperature, normally specified in degrees centigrade,... 

The two variables that can affect the phase equilibria in a one-component, or unary, system are temperature and pressure. The phase diagram for such a system is therefore a temperature-pressure equilibrium diagram. 

Equations (1.6-10) ate the general lelationships for Unary solid-liquid phase diagrams. They d nerate to the case disrassed earlier (no mutual solubilities in the solid phases) on setting 7fzi > 7fz2 = 1. 

Different polymorphs are just different crystalline forms of the same chemical compound having same chemical formula. Thus, they are all represented on the pure compound side of the phase diagram (Figure 3.21b). ContrarUy, solvates refer to the binary system of a compound and a solvent and, therefore, are represented as intermediate compounds between the pure compound and the solvent in the phase diagram (Figure 3.21a). Hence, they differ in chemical formula. Each solvate may have own polymorphic forms (of same chemical formula), which then again belong to the unary system of the intermediate compound. 

Figure 1.4 Phase diagrams for a hypothetical unary system in the form of a Type 1 (P versus T) diagram, a Type 2 (P versus V) diagram and a Type 3 (S versus V) diagram.

Equation (1.72) is the unary Gibbs phase rule. It indicates that the maximum number of phases which can coexist in a unary system is 3 and this results in an invariant equilibrium (f = 0). Note that the equilibria in each type of phase diagram in Figure 1.4 satisfy this condition. 

Figure 1.8 presents the phase equilibria in a hypothetical binary eutectic system similar to that in Figure 1.7, represented on each of the three types of diagrams. This diagram is similar to those for the Ag-Cu and Ni-Cr systems. The plot of T versus ub is a Type 1 diagram and the three-phase equilibrium a-L-(3 is represented by a point. The plot of T versus Ab is a Type 2 diagram and the a-L-(3 equilibrium is represented by three points on a line, the eutectic isotherm. The plot of S versus Xb is a Type 3 diagram and the a-L-(3 equilibrium is represented by an area. Note that the forms of these diagrams correspond to those for the unary system in Figure 1.4. (Numerous examples of the three types of phase diagrams are given for unary, binary and ternary systems in Chapter 13 of Reference [2], Reference [5] and Chapter 2 of Reference [8].

Furthermore, the behavior of the two relevant unary systems (surfactant and water) must be known before the physical science of binary systems can be fully understood [17], C12MG was studied in depth following these principles, so we begin with a description of the equilibrium and kinetic aspects of the phase behavior of the unary (dry) C12MG system. Then these same aspects of the binary C12MG water system will be considered. The binary phase diagram that resulted from these studies has been pubUshed [18]. 

FIG. 7 The unary C12MG phase diagram manifold. The temperature scale is to the left (outside the diagram). Each arm corresponds to a particular crystal structure the left arm is the equilibrium diagram. The transformations that occur among polymorphs are indicated by dashed arrows. 

---