Phase Equilibria, Phase Diagrams and Phase Transformations

Transcription

1 Mats Hillert Department of Materials Science and Engineering KTH, Stockholm Phase Equilibria, Phase Diagrams and Phase Transformations Their Thermodynamic Basis CAMBRIDGE UNIVERSITY PRESS

2 Contents Preface xiii Basic concepts of thermodynamics External State variables Internal State variables The first law of thermodynamics Freezing-in conditions Reversible and irreversible processes The second law of thermodynamics Condition of internal equilibrium Driving force The combined first and second law General conditions of equilibrium Characteristic State functions Entropy 33 V4 Manipulation of thermodynamic quantities Evaluation of one characteristic State function from another Internal variables at equilibrium Equations of State Experimental conditions Notation for partial derivatives 47

3 VIII Contents 2.6 Use ofvarious derivatives Comparison between C v and C P Change of independent variables Maxwell relations 56 Systems with variable composition Chemical potential Molar quantities More about characteristic State functions Various forms of the combined law Partial quantities Relations for partial quantities Alternative variables for composition The lever rule The tie-line rule Different sets of components Constitution and constituents Chemical potentials in a phase with sublattices 88 Evaluation and use of driving force Irreversible thermodynamics Calculation of equilibrium Evaluation of the driving force Evaluation of integrated driving force as function of T or P Driving force for molecular reactions Onsager's reciprocity relations Driving force and entropy production in diffusion Effective driving force 106 Stability Introduction Some necessary conditions of stability Sufficient conditions of stability Summary of stability conditions Limit of stability Limit of stability ofalloys Chemical capacitance Limit of stability in phases with sublattices Le Chatelier's principle 129 Applications of molar Gibbs energy diagrams Molar Gibbs energy diagrams for binary Systems Instability of binary Solutions 139

4 Contents 6.3 Illustration of the Gibbs-Duhem relation Two-phase equilibria in binary Systems Allotropic phase boundaries Effect of a pressure difference on a two-phase equilibrium Driving force for the formation of a new phase Partitionless transformation under local equilibrium Activation energy for a fluctuation Ternary Systems Solubility product 163 Phase equilibria and potential phase diagrams 167 o Q 7.1 Gibbs'phase rule Fundamental property diagram Topology of potential phase diagrams Potential phase diagrams in binary and multinary Systems Sections of potential phase diagrams Binary Systems Ternary Systems Direction of phase fields in potential phase diagrams Extremum in temperature and pressure 199 Molar phase diagrams Molar axes Sets ofconjugatepairscontaining molar variables Phase boundaries Sections of molar phase diagrams Schreinemakers'rule Topology ofsectioned molar diagrams 222 Projected and mixed phase diagrams Schreinemakers'projection of potential phase diagrams The phase field rule and projected diagrams Relation between molar diagrams and Schreinemakers' projected diagrams Coincidence of projected surfaces Projection of higher-order invariant equilibria The phase field rule and mixed diagrams Selection of axes in mixed diagrams Konovalov's rule General rule for singular equilibria 255 Direction of phase boundaries Useof distribution coefficient 258

5 X Contents 10.2 Calculation of allotropic phase boundaries Variation of a chemical potential in a two-phase field Direction of phase boundaries Congruent melting points Vertical phase boundaries Slope of phase boundaries in isothermal sections The effect of a pressure difference between two phases 281 Sharp and gradual phase transformations Experimental conditions Characterization of phase transformations Microstructural character Phase transformations in alloys Classification of sharp phase transformations Applications of Schreinemakers' projection Scheil's reaction diagram Gradual phase transformations at fixed composition Phase transformations controlled by a chemical potential 308 Transformations at constant composition The phase field rule at constant composition Reaction coefficients in sharp transformations for p = c Graphical evaluation of reaction coefficients Reaction coefficients in gradual transformations for p = c Driving force for sharp phase transformations Driving force under constant chemical potential Reaction coefficients at a constant chemical potential Compositional degeneracies for p = c Effect to two or more compositional degeneracies for p = c Compositional degeneracies and the phase field rule Overlapping transformations 343 Partitionless transformations Deviation from local equilibrium Adiabatic phase transformation Quasi-adiabatic phase transformation Partitionless transformations in binary System Partial chemical equilibrium Transformations in steel under quasi-paraequilibrium Transformations in steel under partitioning of alloying elements 366 Limit of stability, critical phenomena and interfaces Transformations and transitions 368

6 Contents XI ^ Q 14.2 Order-disorder transitions Miscibility gaps Spinodal decomposition Tri-critical points Interfaces Nucleation 395 Methods of modelling General principles Choice of characteristic State function Reference states Representation ofgibbs energy offormation Use of power series in T Representation of pressure dependence Application of physical modeis Ideal gas Real gases Mixtures of gas species Black-body radiation Electron gas 425 Modelling of disorder Introduction Thermal vacancies in a crystal Topological disorder Heat capacity due to thermal vibrations Relation between vibrational energy and elastic properties Magnetic contribution to thermodynamic properties A simple physical model for the magnetic contribution The physical background ofmagnetism Random mixture ofatoms Restricted random mixture Crystals with stoichiometric vacancies Interstitial Solutions 455 Mathematical modelling of Solution phases Ideal Solution Mixing quantities Excess quantities Empirical approach to substitutional Solutions Real Solutions Applications of the Gibbs-Duhem relation Dilute Solution approximations 476

7 XII Contents 17.8 Predictions for Solutions in higher-order Systems Numerical methods of predictions for higher-order Systems 482 Solution phases with sublattices Sublattice Solution phases Interstitial Solutions Reciprocal Solution phases Combination of interstitial and substitutional Solution Phases with variable order Ionic solid Solutions 499 Physical Solution modeis Concept of nearest-neighbour bond energies Random mixing model for a substitutional Solution Deviation from random distribution Short-range order Long-range order Long-and short-range order The Compound energy model with short-range order Interstitial ordering Composition dependence of physical effects 522 References 527 Index 529