Muyu Zhao

The Application of the Boundary Theory to Isobaric Phase Diagrams

Through application of the boundary theory, Palatnik-Landau’s contact rule for phase regions is deduced. Then, and in accord with the course of deduction of this rule, its limitations in some of it application have been examined. Comparison of the Palatnik-Landau’s contact rule with the boundary theory, has been discussed in detail.

The Phase Rule, Its Deduction and Application

The Gibbs phase rule is now a long established principle of Chemistry, very well known by all physical chemists and materials scientists. So, why do we still need to write a chapter to discuss this classic, fundamental law of Chemical science, at the outset of this treatise?

The Boundary Theory of Multicomponent Isobaric Isopleth Sections

When the temperature is varying, the equilibrium phase transitions within sys- tem of three or more components, at constant pressure, may be described by means of isobaric isopleth (or vertical) sections. Thus, isopleths have important application over a great range of the different disciplines within the various sciences and technologies. In applying the boundary theory to isopleth section, the relationship among the NPRs and their boundaries in the isobaric isopleths may be systematically clarified.

The Boundary Theory for p - T - x i Multicomponent Phase Diagrams

The multicomponent phase equilibrium, in which p-T-x i are independent variables, is one of the key problems studied in geology, high-pressure physics and high-pressure chemistry. The equilibrium, as applied to geology, are usually presented in p-T phase diagrams, with invariant composition parameters or temperature-composition phase diagrams, set at different given pressures. We now discuss the p-T-x i multicomponent phase diagrams in this text; such analyses may be more comprehensive, and if so, then they are useful for the researches in geology and applied high-pressure physics. When compared with “isobaric” phase diagrams, there is one additional parameter to be considered. Even for the p-T-x binary phase diagrams, they must be depicted into three-dimensional space, therefore, the problem becomes much more complicated.

The Calculation of Unary High-Pressure Phase Diagrams and the Boundary Theory of p - T Phase Diagrams of Multicomponent Systems

The determination of high-pressure phase diagrams, especially those of the multicomponent phase diagram type, is generally quite difficult. Up until now, most experimentally determined phase diagrams at high-pressure are unary in type. The number of published, high-pressure phase diagrams covering binary alloy systems has been rather small. COMPOUND AND ALLOY UNDER HIGH PRESSURE: A HANDBOOK [Tonkov, 1998] has presented information on about 890 binary systems, along with some data on the behavior of 1153 “pseudo” binary and ternary systems, “revealed” up to 1995. This book is the most complete reference work concerning the behavior and related data on alloys exposed to high pressures up till now. Besides the alloy systems, there are also many experimental results now available on the high-pressure, phase equilibrium of other multicomponent systems, mostly on the oxide systems as studied in geology etc.

The Calculation of High-Pressure Ternary Phase Diagrams

The calculation of high-pressure, ternary phase diagrams is a more complicated process than that required for binary systems. Many investigators, especially geologists, have tried to calculate these phase diagrams theoretically. In geology, the calculated results on high-pressure phase equilibrium of multicomponent systems, usually of oxide systems are expressed, in most cases, in p-T and T-x i diagrams. In our studies, we have tried to calculate the high-pressure, multicomponent phase diagrams for alloy systems. Additionally, the principles of our calculation method can be applied to other systems.

Calculation of Binary High-Pressure Phase Diagrams

The basic principle utilized for the calculation of phase diagrams at high pressures is: when a closed system is at equilibrium at the prescribed temperature and pressure, its total Gibbs free energy reaches the minimum value, i:e.

The Application of the Boundary Theory of Phase Diagrams to the Quaternary and Higher Number Component Phase Diagrams

In the text books dedicated to phase diagrams, isobaric phase diagrams are presented primarily as unary, binary and ternary phase diagrams. In the case of displays of two dimensional planes, the quaternary and higher component multidimensional phase diagrams can only be presented as sections or projections. The isobaric, multicomponent horizontal and vertical sections are the particular sections of importance. In order to discuss sections of these types, the boundary theory of phase diagrams is helpful. Moreover, higher dimensional phase diagrams can be inferred from lower ones, for example, in the calculation of isobaric ternary sections, one usually starts from the “side lines” of the binary systems; in these cases, the application of boundary theory is very enlightening. For the construction, application or learning/teaching of quaternary or more component system phase diagrams, the boundary theory for these phase diagrams is both useful and indispensable, since most people are usually not familiar with such complicated phase systems and their diagrammatic representation.

Application of the Boundary Theory to Unary, Binary and Ternary Phase Diagrams

According to the boundary theory of isobaric phase diagrams, the relationship among NPRs and their boundaries in unary, binary and ternary phase diagrams of all types (including isothermal and isopleth sections), could, with advantage, be systematically clarified. With the aid of boundary theory, Rhines’ ten empirical rules concerning the construction of complicated ternary phase diagrams from phase diagram units are well expounded. A comparison, made between the boundary theory and Palatnik-Landau’s contact rule of phase regions, is also presented in this chapter.

The Boundary Theory of Isobaric Phase Diagrams

A phase diagram is constituted by the phase regions displayed and their boundaries. The relationship among neighboring phase regions (abbreviated to NPRs), and the NPRs’ boundaries is of a certain regularity. In other words, the phase regions and their boundaries must follow certain rules in order to constitute a correct phase diagram. The phase rule, which defines the relation among the number of components, number of phases and degrees of freedom of an equilibrium system, is infallibly applicable in every phase diagram. However, it does not cover the relationship and the regularity among the NPRs and their boundaries in the phase diagrams. For example, it would be difficult to answer the following questions: (1) If the phase assemblages of two NPRs are known, what are the characteristics of the boundaries between them? (2) If the phase assemblage of the first NPR and the characteristic of the boundary between two NPRs are known, what is the phase assemblage of the second NPR?