THE UNARY SYSTEM

T - diagram of silicon dioxide with some of the possible modifications. 

The main modifications of silicon dioxide are a and P quartz, a and P cristobalite and a and P tridymite. 

Conversions between the modifications quartz, tridymite and cristobalite progress only slowly because their crystal structures differ relatively much. The conversion between the a and the 6 form of the same modification on the other hand takes place rather fast because the differences in structure are relatively small. All in all this means that transitions from one modification to another are only possible when they are slowly heated or cooled. This is the only way in which the building blocks are offered the possibility to regroup to form a new crystal structure. By heating or cooling quickly it is possible to skip certain modifications. As you can see in figure 6.6 the transition from a to B quartz at standard pressure takes place at 573 °C. 

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Three kinds of systems can be represented in the composition triangle 1. The vertices on the triangle represent the unary system with the pure components. 

Figure 2.1. Generalized phase relations in the unary system CaCC>3. "A" -aragonite "I" through "V - calcite polymorphs with metastable fields indicated by (). Dash-dot line at 800°C represents transition encountered on cooling runs solid line at lower T represents transition encountered on heating runs. (After Carlson, 1980.)...

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 1.5 A schematic plot of G versus Tat a pressure of 1 atm for the unary system in Figure 1.4.

The application of the Gibbs phase rule to multicomponent systems containing c components may he done by extending the treatment used for the unary system in Section 1.2.1.1. Again let ip represent the number of... 

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].

Notice that the structures presented in this paragraph are unary structures, that is one species only is present in all its atomic positions. In the prototypes listed (and in the chemically unary isostructural substances) this species is represented by a pure element. In a number of cases, however, more than one atomic species may be equally distributed in the various atomic positions. If each atomic site has the same probability of being occupied in a certain percentage by atoms X and Y and all the sites are compositionally equivalent, the unary prototype is still a valid structural reference. In this case, from a chemical point of view, the structure will correspond to a two-component phase. Notice that there can be many binary (or more complex) solid solution phases having for instance the Cu-type or the W-type structures. Such phases are formed in several metallic alloy systems either as terminal or intermediate phases. 

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... 

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... 

For the unary diagram, we only had one component, so that composition was fixed. For the binary diagram, we have three intensive variables (temperature, pressure, and composition), so to make an x-y diagram, we must fix one of the variables. Pressure is normally selected as the fixed variable. Moreover, pressure is typically fixed at 1 atm. This allows us to plot the most commonly manipulated variables in a binary component system temperature and composition. 

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... 

In phase rule systems are categorized according to the number of components unary systems with only one component, binary systems with two components and (in this book) finally ternary systems with three components. The behaviour of the components in a system is determined by variables pressure, temperature and composition. 

Phase changes are effected by three externally controllable variables. These are pressure, temperature and composition. In a one-component system, or unary system, however, the composition does not vary, but must always be unity. Therefore there are only two variables which can vary pressure and temperature. Every possible combination of temperature and pressure can be readily represented by points on a two-dimensional diagram. 

This is not a true unary system, but a binary system of A and the inert gas. In this case, the vapor pressure of A will be adjusted so that the gaseous A at 0.8 atm exists in equilibrium with the liquid A at 7j. The partial pressure of the inert gas will be 0.2 atm to maintain the total pressure of 1 atm. 

The phase rule(s) can be used to distinguish different types of equilibria based on the number of degrees of freedom. For example, in a unary system, an invariant equilibrium (/ = 0) exists between the liquid, solid, and vapor phases at the triple point, where there can be no changes to temperature or pressure without reducing the number of phases in equilibrium. Because / must equal zero or a positive integer, the condensed phase rule (/ = c — p + 1) limits the possible number of phases that can coexist in equilibrium within one-component condensed systems to one or two, which means that other than melting, only allotropic phase transformations are possible. Similarly, in two-component condensed systems, the condensed phase rule restricts the maximum number of phases that can coexist to three, which also corresponds to an invariant equilibrium. However, several invariant reactions are possible, each of which maintains the number of equilibrium phases at three and keeps / equal to zero (L represents a liquid and S, a solid) ... 

Once the unary and binary potentials, which describe interactions in the wall-ion system, are specified, distribution of ions near the wall, and between slabs can be calculated. We will further employ an approximation based upon the distribution function formalism, but first the definitions of equilibrium ion densities and corresponding distribution functions have to be introduced. 

The experimental results referred to above lead to the conclusion that liquids consist of mixtures of different molecular species. In presence of traces of moisture, internal equilibrium between the different molecular species is established rapidly, and the liquid behaves as a unary system, but in absence of moisture, internal equilibrium is not established (or is established very slowly), and the liquid behaves like a multi-component system, the vapour pressure depending not only on the temperature, but also on the amounts of the different molecular species (pseudo-components) in the liquid. 

It has already been indicated (p. 53), that in the case of sulphur we have a substance which can give rise to different molecular species which, in the liquid state, form an equilibrium mixture. For this reason, sulphur will behave not as a purely unary system, but in a manner similar to that of dynamic isomerides discussed in this chapter. We shall therefore discuss briefly the more important equilibrium relations of sulphur from this point of view. 

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. 

Si-B-C-N phase equilibria were calculated by extrapolating from the related unary, binary and ternary systems into quaternary compositions [236, 244, 271-273]. This is valid since no solid solubilities and no quaternary phases exist in the quaternary system. Calculated phase equilibria in the Si-B-C-N system were shown exemplarily by isothermal sections at 1673 K and 2273 K, respectively (Figs. 24a and b). 

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. 

There are no ternary compounds in flie Cu-Fe-Mo system. The unary and binary phases discussed in this assessment are listed in Table 2. 

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