Solidification and Crystallization Processing in Metals and Alloys
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Persons
Hasse Fredriksson, KTH, Sweden.
Ulla ?kerlind, University of Stockholm, Sweden.
Content
Preface ix
1 Thermodynamic Concepts and Relationships 1
1.1 Introduction 2
1.2 Thermodynamic Concepts and Relationships 2
1.3 Thermodynamics of Single-Component Systems 10
1.4 Thermodynamics of Multiple-Component Systems 16
1.5 Thermodynamics of Alloys 21
1.6 Thermodynamics of Ideal Binary Solutions 25
1.7 Thermodynamics of Non-Ideal Binary Solutions 26
1.8 Experimental Determination of Thermodynamic Quantities of Binary Alloys 34
Summary 36
Further Reading 41
2 Thermodynamic Analysis of Solidification Processes in Metals and Alloys 42
2.1 Introduction 42
2.2 Thermodynamics of Pure Metals 43
2.3 Thermodynamics of Binary Alloys 44
2.4 Equilibrium Between Phases in Binary Solutions. Phase Diagrams of Binary Alloys 50
2.5 Driving Force of Solidification in Binary Alloys 76
2.6 Thermodynamics of Ternary Alloys 78
2.7 Thermodynamics of Vacancies in Pure Metals and Alloys 83
Summary 90
Exercises 94
References 98
Further Reading 98
3 Properties of Interfaces 99
3.1 Introduction 99
3.2 Classical Theory of Interface Energy and Surface Tension 100
3.3 Thermodynamics of Interphases 114
3.4 Structures of Interfaces 123
3.5 Equilibrium Shapes of Crystals 145
Summary 155
Exercises 161
References 165
4 Nucleation 166
4.1 Introduction 166
4.2 Homogeneous Nucleation 167
4.3 Heterogeneous Nucleation. Inoculation 179
4.4 Nucleation of Bubbles 190
4.5 Crystal Multiplication 193
Summary 194
Exercises 197
References 200
5 Crystal Growth in Vapours 201
5.1 Introduction 202
5.2 Crystal Morphologies 202
5.3 Chemical Vapour Deposition 203
5.4 Crystal Growth 206
5.5 Normal Crystal Growth of Rough Surfaces in Vapours 210
5.6 Layer Crystal Growth of Smooth Surfaces in Vapours 214
5.7 Influence of Impurities on Crystal Growth in Vapours 238
5.8 Epitaxial Growth 244
5.9 Whisker Growth 249
5.10 Mechanical Restrictions on Thin Films 251
Summary 256
Exercises 263
References 266
6 Crystal Growth in Liquids and Melts 267
6.1 Introduction 268
6.2 Structures of Crystals and Melts 268
6.3 Growth Methods 275
6.4 Crystal Growth 278
6.5 Volume Changes and Relaxation Processes during Anelastic Crystal Growth in Metal Melts 279
6.6 Normal Crystal Growth in Pure Metal Melts 289
6.7 Layer Crystal Growth of Smooth Surfaces in Liquids 297
6.8 Normal Crystal Growth in Binary Alloys 310
6.9 Diffusion-Controlled Growth of Planar Crystals in Binary Alloys 318
6.10 Diffusion-Controlled Growth of Spherical Crystals in Alloys 324
6.11 Impingement 338
6.12 Precipitation of Pores 340
Summary 343
Exercises 355
References 360
7 Heat Transport during Solidification Processes. Thermal Analysis 361
7.1 Introduction 362
7.2 Basic Concepts and Laws of Heat Transport 362
7.3 Convection 366
7.4 Theory of Heat Transport at Unidirectional Solidification 378
7.5 Production of Single Crystals by Unidirectional Solidification 385
7.6 Thermal Analysis 395
7.7 Variable Heat of Fusion of Metals and Alloys 408
7.8 Variable Heat Capacitivity of Metals and Alloys 416
Summary 422
Exercises 429
References 432
8 Crystal Growth Controlled by Heat and Mass Transport 433
8.1 Introduction 433
8.2 Heat and Mass Transports in Alloys during Unidirectional Solidification 434
8.3 Zone Refining 443
8.4 Single Crystal Production by Czochralski Technique 450
8.5 Cellular Growth. Constitutional Undercooling. Interface Stability 454
Summary 465
Exercises 469
References 474
9 Faceted and Dendritic Solidification Structures 475
9.1 Introduction 476
9.2 Formation of Faceted Crystals 476
9.3 Growth of Faceted Crystals in Pure Metal Melts 476
9.4 Growth of Faceted Crystals in Alloy Melts 485
9.5 Growth of Dendrite Crystals 503
9.6 Development of Dendrites 529
9.7 Transitions between Structure Types in Alloys 565
Summary 573
Exercises 580
References 585
10 Eutectic Solidification Structures 587
10.1 Introduction 588
10.2 Classification of Eutectic Structures 588
10.3 Normal Eutectic Growth 589
10.4 Degenerate and Coupled Eutectic Growth 620
10.5 Structures of Ternary Alloys 635
10.6 Solidification of Fe-C Eutectics 646
10.7 Solidification of Al-Si Eutectics 658
10.8 Transition between Normal Lamellar and Rod Eutectic Growth 665
Summary 669
Exercises 675
References 680
11 Peritectic Solidification Structures 681
11.1 Introduction 681
11.2 Peritectic Reactions and Transformations 682
11.3 Peritectic Reactions and Transformations in Iron-Base Alloys 693
11.4 Metastable Reactions in Iron-Base Alloys 702
11.5 Metatectic Reactions and Transformations 704
11.6 Microsegregation in Iron-Base Alloys 708
11.7 Transitions between Peritectic and Eutectic Reactions in Iron-Base Alloys 716
Summary 719
Exercises 724
References 727
12 Metallic Glasses and Amorphous Alloy Melts 729
12.1 Introduction 730
12.2 Basic Concepts and Definitions 731
12.3 Production of Metallic Glasses 735
12.4 Experimental Methods for Structure Determination of Metallic Glasses and Amorphous Alloy Melts 741
12.5 Structures of Metallic Glasses 744
12.6 Comparison of the Structures of Metallic Glasses and Amorphous Alloy Melts. Rough Models of Metallic Glasses and Amorphous Alloy Melts 753
12.7 Casting of Metallic Glasses. Crystallization Processes in Amorphous Alloy Melts 759
12.8 Classification of Metallic Glasses 763
12.9 Properties and Applications of Metallic Glasses 764
Summary 771
Exercises 775
References 779
Answers to Exercises 781
Index 803
Chapter 2
Thermodynamic Analysis of Solidification Processes in Metals and Alloys
2.1 Introduction
Solidification processes in metals and alloys are controlled by kinetic laws and depend on the driving force of the process. In Chapter 1 on page 9 it is mentioned that the change of Gibbs' free energy of a system can be regarded as the driving force of chemical and metallurgical reactions. The driving force of solidification equals the change in Gibbs' free energy when the system is transferred from a liquid to a solid state.
To be able to analyze the solidification process we must primarily find the Gibbs' free energy of liquid and solid metals and alloys as a function of composition and temperature. The tools we need for this purpose are found in Chapter 1.
In this chapter the driving force of solidification of pure metals is derived. A binary system is far more complicated than a single-component system. After an extensive study of the thermodynamics and stability of ideal and non-ideal solutions the driving force of solidification of binary alloys is derived.
Special care has been devoted to the fundamental condition of equilibrium between phases and the relationships between the Gibbs' free energies of liquid and solid binary alloys and the corresponding phase diagrams of the alloys. Ternary alloys are discussed shortly. The chapter ends with a short section on the thermodynamics of lattice defects in metals.
This theoretical basis together with the laws of physics will be applied in the following chapters where different aspects of the solidification of metals and alloys will be studied and analyzed.
2.2 Thermodynamics of Pure Metals
2.2.1 Driving Force of Solidification
The concept of Gibbs' free energy can be used to understand the behaviour of a metal close to its melting point TM.
Figure 2.1 is a combination of Figures 1.7 and 1.8 on page 16 in Chapter 1. It shows how the molar Gibbs' free energies Gm of a pure liquid and of the corresponding solid metal varies as a function of temperature.
Figure 2.1 The molar Gibbs' free energy of a metal as a liquid and a solid as a function of temperature.The stable state is the one with lowest possible free energy.
At the melting point the molar Gibbs' free energies of the melt and the solid are equal. Below the melting point the solid has a lower free energy than the melt and is therefore the stable phase. Above the melting point the reverse is true and the liquid is more stable than the solid.
The difference in molar Gibbs' free energy between the liquid and the solid at the same temperature below the melting point acts as a driving force for solidification of the undercooled liquid. The greater the driving force is, the stronger will be the tendency of crystallization.
During a spontaneous solidification process the Gibbs' free energy change is negative because heat is lost to the surroundings. It is desirable that the driving force is positive for a spontaneous process and for this reason it is defined as
Driving force = the negative change of the Gibbs' free energy at solidification.
(2.1)
In order to derive a useful expression of the driving force of solidification we apply Equation (1.29) on page 9 in Chapter 1 on 1 kmol of the liquid and solid phases and take the differences between these equations. The result is
At the melting point T = TM the liquid and solid phases are in equilibrium with each other, which can be expressed as . Inserting these values into Equation (2.2) we obtain
(2.3)
This expression of is introduced into Equation (2.2):
After rearrangement of Equation (2.4) the driving force can be written
where
= driving force of solidification
TM = melting-point temperature
ΔT = undercooling (TM − T)
= molar heat of fusion .
The driving force of solidification close to the melting point is proportional to the undercooling and the heat of fusion.
2.3 Thermodynamics of Binary Alloys
The molar Gibbs' free energies of ideal and non-ideal binary solutions as functions of their compositions have been treated briefly in Sections 1.6 and 1.7, respectively, in Chapter 1. The topic will be further penetrated here and the functions will be shown graphically.
In Chapter 1 the molar Gibbs' free energies of ideal and non-ideal binary solutions as functions of composition with the temperature as a parameter were derived. Most of the graphical illustrations of the molar Gibbs' free energy of ideal and non-ideal binary solutions in the following sections are based on these functions, i.e. on Equations (1.86) on page 26 and (1.130) on page 32 in Chapter 1.
Molar Gibbs' free energy of an ideal binary solution
Molar Gibbs' free energy of a non-ideal binary solution
The different terms and the possibilities to measure or calculate the excess terms have been discussed in Sections 1.7 and 1.8 in Chapter 1.
2.3.1 Gibbs' Free Energy of Ideal Binary Solutions
The molar Gibbs' free energy of an ideal binary solution is shown in Figure 2.2. It is identical to Figure 1.13a on page 26 in Chapter 1. The dotted line represents the first two terms in Equation (2.6). The third term in Equation (2.6) is negative and equal to () (Chapter 1 on page 25). It causes a negative deviation from the dotted line in Figure 2.2. No excess-energy term is present. Equation (2.6) and Figure 2.2 may refer either to a liquid or to a solid solution.
Figure 2.2 The molar Gibbs' free energy of an ideal binary solution as a function of composition. The composition is given in mole fraction x in the figure. This is the usual unit when Gibbs' free energy is concerned.
The composition of the solution is given in mole fraction of the solute B in the figure. This is the common unit when Gibbs' free energy is concerned.
The molar Gibbs' free energies of pure component A and of pure component B are given by the intersections between the curve and the vertical lines xB = 0 and xB = 1. Superscript “0” indicates that the elements A and B are at their standard states. and vary with temperature according to Equations (1.58) and (1′) and Figures 1.7 and 1.8 are four on page 15 in Chapter 1. The standard state values are constant at a fixed temperature.
The whole Gm curve lies below the dashed line, which means that a solution of any composition of components A and B is stable, in agreement with the definition of an ideal solution. Every solution, independent of composition, has lower free energy than the unmixed components. Hence, a spontaneous mixing will occur when the components are brought together.
The dissolving process can be followed stepwise in Figure 2.3. Initially, we have a mixture of two components A and B in the proportions xA = 1 − x and xB = x. The initial molar free energy is , found at the intersection between the dashed line and the vertical line xB = x.
Figure 2.3 Dissolution process resulting in a homogeneous single phase.The molar Gibbs' free energy as a function of composition expressed as molar fraction.
When the components start to dissolve, the concentrations near the components A and B changes gradually. This is described by two sliding points on the curve, moving from P0 to P1 to P2, respectively, Q0 to Q1 to Q2. Simultaneously, the molar free energy decreases from to G1 to G2 and finally to its lowest value G3.
The final points P3 and Q3 coincide and correspond to a single homogeneous stable phase. The line through the points P3 and Q3 is the tangent to the curve at the point P3 = Q3. The molar Gibbs' free energy at point P3 is G3. Other ways along the curve can occur, depending on how the dissolution process proceeds in detail, but the end will always be the same (P3 = Q3 and G3).
2.3.2 Gibbs' Free Energy of Non-Ideal Solutions
The molar excess Gibbs' free energy is the difference between the molar Gibbs' free energy of the real non-ideal solution and the one of an imaginary ideal solution of the same set of components.
Equation(2.8) can be written with the aid of Equation (2.6)
Excess quantities of non-ideal solutions were discussed in Chapter 1 (Section 1.7.2 on page 30). The term in Equation (2.9) has to be replaced by quantities, which can be calculated, or measured. Using the basic relationship G = H − TS on the excess functions we obtain
According to Equation (1.114) on page 31 in Chapter 1 we have
The remaining problem...
