The Boundary Theory of Phase Diagrams and Its Application : Rules for Phase Diagram Construction with Phase Regions and Their Boundaries

The Boundary Theory of Phase Diagrams and Its Application -- Rules for Phase Diagram Construction with Phase Regions and Their Boundaries presents a novel theory of phase diagrams. Thoroughly revised on the basis of the Chinese edition and rigorously reviewed, this book inspects the general feature and structure of phase diagrams, and reveals that there exist actually two categories of boundaries. This innovative boundary theory has solved many difficulties in understanding phase diagrams, and also finds its application in constructing multi-component phase diagrams or in calculating high-pressure phase diagrams. Researchers and engineers as well as graduate students in the areas of chemistry, metallurgy and materials science will benefit from this book. Prof. Muyu Zhao was the recipient of the 1998 Prize for Progress in Science and Technology (for his work on the boundary theory of phase diagrams) awarded by the National Commission of Education, China, and many other prizes. Title Page Copyright Page Preface Comment Introduction Table of Contents Part One The Phase Rule, Its Deductionand Application Chapter 1 The Phase Rule, Its Deduction andApplication 1.1 Why do We Discuss the Phase Rule at First 1.2 Different Methods for Deducing the Phase Rule: The Method of Gibbs Himself, Gibbs-Roozeboom’s Methodand the Method of Gibbs Free Energy Minimization 1.2.1 The deduction of the phase rule in the circumstances withoutchemial reaction 1.2.1.1 Gibbs’ method [Gibbs, 1950] 1.2.1.2 Gibbs-Roozeboom’s method 1.2.2 The determinatin of the phase rule involving the circumstanes of chemical reactions, the method of the Gibbs free energy minimization 1.2.2.1 The chemical formulas for both the independent components and the derivedcomponents and the relation existing between them 1.3 Determination of the Number of Independent Componentsby Brinkley’s Method 1.3.1 Brinkley’s method 1.3.2 The relation between the Brinkley’s method and the Jouguet’smethods [Zhao et al., 1992] 1.3.3 The strengths and shortcomings of both the Brinkley’s methodand the Jouguet’s method [Zhao et al, 1992] 1.3.4 Examples 1.4 Some Remarks on the Application of the Phase Rule References-1 Summary of Part One Part Two The Boundary Theory of Isobaric PhaseDiagrams and Its Application Chapter 2 The Boundary Theory of Isobaric Phase Diagrams 2.1 Introduction 2.2 Several Basic Concepts for Underlying the Phase Diagram 2.2.1 Coordinate axes 2.2.2 The phase point and the system point 2.2.3 The isobaric phase diagram and its dimensions 2.2.4 The phase, the phase region and the number of phases existing in the phase region 2.2.5 Neighboring phase regions (abbreviated to NPRs) and the total number of all the different phases present in them, Φ 2.2.6 The maximum phase number in any phase region of the two ormore NPRs, φmax 2.2.7 The boundary and the phase boundary among two or more NPRs 2.2.7.1 Boundary 2.2.7.2 Phase boundary 2.2.7.3 The relationship among the NPRs and their boundaries 2.2.8 More on the phase boundary concept 2.3 The Theorem of the Corresponding Relationship between the Total Number of All the Different Phases in NPRs Φ and the Dimensions of the Phase Boundary R1 in Phase Diagrams, and Its Theoretical Deduction 2.3.1 The theorem of the corresponding relationship between Φ and R1 2.3.2 The theoretical deduction of TCR 2.4 The Theorem of the Corresponding Relationship (TCR)is an Independent Theorem, Not a Variant of the PhaseRule 2.5 Corollaries of TCR for Isobaric Phase Diagrams 2.6 The Relationship between the Dimensions of the PhaseBoundary R1 and the Dimensions of the Boundary R1' for Isobaric Multicomponent Phase Diagrams 2.6.1 The qualitative explanation of the two formulas between R1' and R1 2.6.2 The theoretical proof of the two formulae between R1 and R1' in N > 2 isobaric phase diagrams 2.6.2.1 Two types of phase transitions in the phase diagram 2.6.2.2 A special case of the phase region transfer 2.6.2.3 The theoretical proof of eq. (2-13), R1 =R1 + φC − 1 2.6.2.4 The theoretical proof of R1 =0, there is an invariant phase transition betweentwo NPRs and R1' = R1 + φC 2.7 The Summary of the Boundary Theory of Isobaric Phase Diagrams References-2 Chapter 3 Application of the Boundary Theory to Unary, Binary and Ternary Phase Diagrams 3.1 Determination of Phase Assemblages of NPRs and the Characteristics of Their Boundaries by the Boundary Theory 3.1.1 Determination of the phase assemblage of the second NPR, whenthe phase assemblage of the first NPR, and the characteristics ofthe boundary between the two NPRs, are known 3.1.2 The determination of the characteristics of the boundary betweenthe two NPRs 3.1.3 Examples 3.2 Application of the Boundary Theory to Unary Phase Diagrams 3.3 Application of the Boundary Theory to Binary Phase Diagrams 3.3.1 The general analyses of isobaric binary phase diagrams 3.3.2 Analyses of some typical isobaric binary phase diagrams 3.3.2.1 A typical isobaric binary phase diagram 3.3.2.2 SiO2-Al2O3 phase diagram 3.3.2.3 The phase diagram of the system, KCl-NaCl, with a melting point minimum 3.3.3 A few cases are beyond the scope of the boundary theory 3.3.3.1 The line Mcde in the SiO2-Al2O3 binary phase diagram (Fig. 3.3) 3.3.3.2 The case of the phase region II (L+S), is shown in Fig. 3.4 to be passingto the phase region III (L+S), through the point M, as indicated in Fig. 3.4. 3.3.4 Critical point 3.4 Application of the Boundary Theory to Ternary Phase Diagrams 3.4.1 The general analysis 3.4.2 Isothermal sections of isobaric, ternary phase diagrams 3.4.3 The two types of boundaries 3.4.3.1 The boundary line 3.4.3.2 The boundary point 3.4.4 Typical isopleth sections (or vertical sections) of is obaric ternary phase diagrams 3.4.4.1 A brief analysis of the isobaric ternary isopleth section 3.4.4.2 The characteristics of the boundary lines on a typical, regular isobaric ternary isopleth section 3.4.4.3 The characteristics of the boundary points in the regular, isobaric ternary isopleth section 3.4.5 The boundary theory of isobaric ternary isopleth sections 3.4.5.1 The basic principles 3.4.5.2 The case of R1 > 1 3.4.5.3 The case of the isobaric, ternary spatial phase diagram, R1=0; and two NPRs exist on the same side, above the invariant phase transition temperature.There is no (3+1) = 4 four-phase coexisting invariant region between the two NPRs,then R1 = R1 + φC − 1 3.4.5.4 The case: in the isobaric ternary phase diagram, R1=0; an invariant phasetransition from one NPR to another exists, and there is a (N+1)= 4 four-phaseinvariant region, situated between the two NPRs, so R1 = R1 + φC 3.4.5.5 The analysis of regular and irregular isopleth sections with the boundary theory 3.5 Explanation of Rhines’ Ten Empirical Rules for Constructing Complicated Ternary Phase Diagrams with the Bounary Theory 3.6 Comparison of the Boundary Theory and the P-L's ContactRule of Phase Regions 3.6.1 The deduction of contact rules of phase regions by applying the boundary theory 3.6.2 The meanings of the parameters used in the boundary theory are clearer than those used in the Palatnik-Landau’s theory 3.6.3 The difficulties of applying the contact rule 3.6.4 The merits of the boundary theory References-3 Chapter 4 The Application of the Boundary Theory of Phase Diagrams to the Quaternary and Higher Number Component Phase Diagrams 4.1 Introduction 4.2 The Relationship among NPRs and their Boundaries in a Typical, Isobaric, Quaternary Phase Diagram 4.3 During Temperature Decreasing, Some Cases of Variations of the NPRs and their Boundaries, May be Encounteredfor Several Types of Quaternary Isobaric PhaseDiagrams 4.3.1 The simple quaternary phase diagram, in which, (a) the liquidstate components are completely miscible, but (b) the same components are completely insoluble with one another, in the solidstate 4.3.2 The isobaric quaternary phase diagram, in which the 4 componentsin the liquid state are completely miscible, but, in the solidstate, they are partially soluble in one another 4.3.2.1 The boundary lines 4.3.2.2 Boundary points 4.3.3 Isobaric quaternary phase diagrams with peritectic transitions 4.3.3.1 Boundary lines 4.3.3.2 Boundary points 4.3.4 Quaternary phase diagrams, in which the systems have either compoundsor intermediate phases or maximum (or minimum) melting points 4.4 The Isobaric Quinary Phase Diagrams 4.5 Conclusion References-4 Chapter 5 The Boundary Theory in Construction of Multicomponent Isothermal Sections 5.1 The Relationship among Neighboring Phase Regions(NPRs) and Their Boundaries in Isobaric Isothermal Multicomponent Sections 5.1.1 General rules 5.1.2 The two types of boundaries in isobaric is othermal multicomponent sections 5.1.2.1 The boundary lines 5.1.2.2 The boundary point 5.1.2.3 In case Φ − φC > 3 5.2 The Non-Contact Phase Regions and the Boundaries between Them 5.2.1 The number of boundaries existing between non-contact phase regions 5.2.2 The course takes a zigzag path 5.3 Construction of an Isothermal Quinary Section, with Limited Information 5.3.1 The method of the boundary theory 5.3.2 Gupta’s method for constructing multi-component isothermal sections[Gupta et al., 1986] 5.4 The Method of Constructing an Isothermal Eight-Component Section 5.5 Summary of Using the Boundary Theory Method References-5 Chapter 6 The Boundary Theory of Multicomponent Isobaric Isopleth Sections 6.1 Introduction 6.1.1 General rules 6.2 The Characteristics of Boundaries in Isopleth Sectionsfor the Case of N > 2 and R1 > 1 6.2.1 N > 2 and R1 > 1, there is boundary line (R>1)i = 1 in the isopleth section 6.2.2 N > 2, R1 > 1, and there are boundary points (R>1)i = 0 in theisopleth section 6.2.2.1 The case (1): the boundary point is also a phase boundary point in theisopleth section. 6.2.2.2 The case (2): in the isopleth section, there are boundary points, but notphase boundary points, i.e. 0=(R1)i > (R1)i. 6.3 The Characteristics of Boundaries in the Isopleth Sectionfor the Case of N  2, R1 = 0, there is no Invariant Phase Transition between the two NPRs 6.3.1 There is no boundary line between the two NPRs for the above mentioned case 6.3.2 The boundary points with (R>1)i = 0 6.4 The Case of N > 2, R1 = 0, there is an Invariant PhaseTransition between the two NPRs. In this Case, theremay be a Boundary Line or a Boundary Point between two NPRs 6.4.1 In the isopleth section, boundary lines exist: (R1)i = 1 betweenthe NPRs in the above condition 6.4.2 The boundary points (R1)i = 0 between the two NPRs in the case mentioned above 6.5 Example 6.6 The Theory of Two-Dimensional Sections of Isobaric Multicomponent Phase Diagrams References-6 Chapter 7 The Application of the Boundary Theory to Isobaric Phase Diagrams 7.1 Brief Review of the Application for the Boundary Theory 7.2 The Analysis of the Fe-Cr-C Isopleth Section 7.3 The Application of the Boundary Theory to Phase Diagram Calculation 7.3.1 The general principles are now discussed for the phase diagram calculation with the aid of the boundary theory 7.3.2 Direct calculation of isopleth sections of the Bi-Sn-Zn ternary system 7.3.2.1 The calculation of a regular section 7.3.2.2 Calculation of an irregular isopleth section 7.3.2.3 Further discussions 7.4 Application of the Boundary Theory to Phase Diagram Assessment 7.4.1 In-Zr binary phase diagram 7.4.2 Phase diagrams of rare-earth metals 7.5 Application of the Boundary Theory to Phase Diagram Determination 7.6 The Application of the Boundary Theory to Phase Diagram Education References-7 Summary of Part Two Part Three The Boundary Theory and Calculation of High Pressure Phase Diagrams Chapter 8 The Boundary Theory for p-T-xi Multicomponent Phase Diagrams 8.1 Introduction 8.2 The Theorem of Corresponding Relationship for p-T-xi Multicomponent Phase Diagrams and Its Corollaries 8.2.1 The theorem of corresponding relationship 8.2.2 The corollaries of TCR 8.3 The Relationship between R'1 and R1 in p-T-xi MulticomponentPhase Diagrams [Zhao, 1985] 8.4 The Relationship among NPRs and Their Boundaries for the p-T-x Binary Phase Diagrams. 8.4.1 A simple case 8.4.2 Complex p-T-x binary phase diagrams 8.5 Relationship among NPRs and their Boundaries for thep-T-xi Ternary Phase Diagram 8.5.1 A simple case 8.5.2 Complicated p-T-xi ternary phase diagrams 8.5.3 Some remarks 8.6 The Application of Boundary Theory for Quaternary p-T-xi Phase Diagrams 8.7 The Reliability of the Boundary Theory of Multicomponentp-T-xi Phase Diagrams References-8 Chapter 9 The Calculation of Unary High-Pressure Phase Diagrams and the BoundaryTheory of p-T Phase Diagrams of Multicomponent Systems 9.1 Introduction 9.2 Calculation of Unary p-T Diagrams 9.3 The Boundary Theory of p-T Phase Diagrams of Multicomponent Systems without Composition Variable References-9 Chapter 10 Calculation of Binary High-Pressure Phase Diagrams 10.1 Principles for the Calculation of Binary Phase Diagrams at Elevated Pressures 10.2 Calculation of the Standard Molar Gibbs Free Energy for the Pure Components 10.3 Calculation of Activity Coefficients γi(T, p0, xi) of thei-th Component in the Equilibrium Phases 10.3.1 The activity coefficient γi(T, p0, xi) of the i-th component in the liquid phase 10.3.2 Activity coefficient γiS(T, p0, xiS) of the i-th component in the solid phase 10.4 Partial Molar Volumes 10.4.1 Partial molar volume of the i-th component in the liquid phase, ̄ ViL(T, p, xiL) 10.4.2 Partial molar volume of the i-th component in the solid phase ̄ ViS(T, p, xiSj ) 10.5 Some Remarks on the Values of α and β 10.5.1 The coefficient of thermal expansion, αij 10.5.2 Compressibility coefficients of pure components, βij 10.6 Example-Calculation of the Cd-Pb Phase Diagram at High Pressure [Zhou et al., 1990] 10.6.1 The treatment of thermodynamic quantities 10.6.1.1 The treatment of Δμ0i(S→L) and KTi 10.6.1.2 The activity coefficients of the components, γij(T, p0, xij) 10.6.1.3 The volume terms 10.6.2 Calculated results and discussions 10.6.2.1 The effect of pressure on eutectic events 10.6.2.2 Comparison of the calculated and experimental phase diagrams 10.6.2.3 p-x phase diagram of the Cd-Pb system at a given temperature References-10 Chapter 11 The Calculation of High-Pressure Ternary Phase Diagrams 11.1 The Characteristics of the Boundaries of the High-Pressure Ternary Phase Diagrams, and the Basic Equationsfor Their Calculation 11.1.1 The characteristics of the boundaries between NPRs of highpressureternary phase diagrams 11.1.1.1 If (R1)s = (R1 )s, this boundary is not only a boundary consisting of systempoints, it also acts as a phase boundary. 11.1.1.2 If (R1)s = (R1 )s then these boundaries are boundaries only and are not phase boundaries 11.1.2 The basic equations for the calculation of different kinds of boundaries 11.1.2.1 The calculation of phase boundaries (or the boundaries of the first type) 11.1.2.2 The calculation of the boundaries only (or the boundaries of the second type) 11.2 The Treatment of Thermodynamic Parameters for Ternary Systems at High Pressure 11.3 Verification of the Estimation Method for the Excess Molar Volume by Experiment 11.4 The Calculation of High-Pressure Phase Diagrams of Cd-Pb-Sn and Cd-Sn-Zn Systems 11.4.1 Calculation of KTi(T, p0) = exp−μ0i,L − μ0i,SRT  11.4.2 The activity coefficients γi,j(T, p0, xi,j) of the i-th component in the equilibrium phases 11.4.2.1 The activity coefficient γi,j(T, p0, xi,j) (i=1, 2, 3) of the i-th componentin the liquid phase 11.4.2.2 The activity coefficient γi,Sj of the i-th component in solid phase Sj 11.4.3 The partial molar volumes for the Cd-Pb-Sn ternary systems 11.4.3.1 The partial molar volume of the i-th component in the liquid phase for the ternary system 11.4.3.2 The partial molar volume ̄ ViS(T, p, xiS) of the i-th component in the solid phase for the ternary system. 11.5 Verification of Calculated High-Pressure Ternary Phase Diagrams through Experimental Determination 11.6 The Comparison between the Methods of Experimental Determination and Thermodynamic Calculation of High Pressure Phase Diagrams [Song et al., 1993] References-11 Summary of Part Three References of This Book Important Symbols Index Annex