TERNARY SYSTEMS. x 1 + x 2 + x 3 = 1 in the same way: h 1 + h 2 + h 3 = h triangle. Concentration given in Gibbs Triangle. h triangle. h 2. h 3.
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1 TERNARY SYSTEMS Concentration given in Gibbs Triangle x 1 h triangle h 2 h 1 x 3 h 3 x 2 x 3 x 1 x 1 + x 2 + x 3 = 1 in the same way: h 1 + h 2 + h 3 = h triangle VO Phasendiagramme Ternäre Systeme Folie 1
2 TERNARY SYSTEMS Tie Lines S: 20 % A + 70 % B + 10 % C L: 40 % A + 30 % B + 30 % C Mix one part of S with three parts of L Composition of P (must be on the line S-L): Line S-L is a tie line! Lever Rule can be applied! VO Phasendiagramme Ternäre Systeme Folie 2
3 Isomorphous Ternary Systems Space Diagram and possible representations binary tie lines Horizontal Sections: Isotherms Vertical Sections: Isopleths Liquidus Projections VO Phasendiagramme Ternäre Systeme Folie 3
4 Isomorphous Ternary Systems Isothermal Sections Binary limiting tie line Solidus Tie lines in ternary composition range cannot be predicted, but must be determined experimentally! VO Phasendiagramme Ternäre Systeme Folie 4
5 Isomorphous Ternary Systems Vertical Sections = Isopleths Tie lines cannot be shown as they are usually not within this plane VO Phasendiagramme Ternäre Systeme Folie 5
6 Isomorphous Ternary Systems Liquidus Projection Of course, no information possible about tie lines! Gibbs Energy Surface VO Phasendiagramme Ternäre Systeme Folie 6
7 Isomorphous Ternary Systems F = C P + 2 Single-phase field: F = = 4 for p = const F = = 3 Application of Phase Rule Two-phase field (for p = const): F = = 2 select temperature select composition of A in the solid phase Two-phase field (for p = const): F = = 2 select composition of B in the liquid phase select composition of C in the liquid phase VO Phasendiagramme Ternäre Systeme Folie 7
8 Isomorphous Ternary Systems Solidification B C All tie lines have to cross the point of the gross composition: X = L 1 = α VO Phasendiagramme Ternäre Systeme Folie 8
9 Isomorphous Ternary Systems Maxima and Minima Maximum in the binary System A-B Tie Lines: α α L Maximum in the ternary System Tie Lines: α L VO Phasendiagramme Ternäre Systeme Folie 9
10 Ternary Three-Phase Equilibrium F = = 2 for p = constant: F = = 1 if T selected, no degree of freedom left anymore! Tie Triangle P is mixture of 2 parts of R, 3 parts of S, 5 parts of L Ternary Lever Rule VO Phasendiagramme Ternäre Systeme Folie 10
11 Ternary Three-Phase Equilibrium Space Model Three-phase equilibria in ternary systems occur over a temperature range! Example: Bi-Pb-Sb VO Phasendiagramme Ternäre Systeme Folie 11
12 Ternary Three-Phase Equilibrium T 1 T 2 -T 4 T 5 Binary eutectic L α + β L Tie lines (α + β) β Tie lines (L + β) Tie lines (L + α) α Binary eutectic L α + β VO Phasendiagramme Ternäre Systeme Folie 12
13 Ternary Three-Phase Equilibrium Development of Isotherms T 1 T 2 -T 4 T VO Phasendiagramme Ternäre Systeme Folie 13
14 Ternary Three-Phase Equilibrium Possible and forbidden phase boundaries Could be demonstrated by use of Gibbs energy surfaces! VO Phasendiagramme Ternäre Systeme Folie 14
15 Ternary Three-Phase Equilibrium Freezing of an alloy with the composition X Solidification L X Primary crystallization of α begins Secondary crystallization of (α+β) begins Final product: mixture of (α+β) VO Phasendiagramme Ternäre Systeme Folie 15
16 Ternary Three-Phase Equilibrium Space Model two peritectic systems VO Phasendiagramme Ternäre Systeme Folie 16
17 Ternary Three-Phase Equilibrium Space Model various types Termination of a threephase equilibrium in a critical tie line VO Phasendiagramme Ternäre Systeme Folie 17
18 Ternary Three-Phase Equilibrium Critical Points and Critical Curves critical points critical curve Participation of a binary monotectic critical tie line critical curves VO Phasendiagramme Ternäre Systeme Folie 18
19 Ternary Three-Phase Equilibrium Maxima and Minima The three-phase equilibrium may pass through a maximum or minimum A (α+β+l) B A B C Example of a maximum C A B A B C VO Phasendiagramme Ternäre Systeme Folie 19 C
20 Ternary Three-Phase Equilibrium Vertical Sections VO Phasendiagramme Ternäre Systeme Folie 20
21 Ternary Three-Phase Equilibrium Ternary Diffusion Couples VO Phasendiagramme Ternäre Systeme Folie 21
22 Ternary Four-Phase Equilibria Three different types of four-phase equilibria Class I (or E-type): L α + β + γ (ternary eutectic) Class II (or U-Type): L + α β + γ (transition reaction) Class III (or P-type): L + α + β γ (ternary peritectic) Four phase equilibria are invariant: F = C P + 2 = = 1 with p = const. F = VO Phasendiagramme Ternäre Systeme Folie 22
23 Ternary Four-Phase Equilibrium L α + β + γ Class I A e 1 I (E) B e 3 Schematic Liquidus Projection e 2 C VO Phasendiagramme Ternäre Systeme Folie 23
24 Ternary Four- Phase Equilibrium Class I VO Phasendiagramme Ternäre Systeme Folie 24
25 Ternary Four-Phase Equilibrium Class I Isothermal Sections VO Phasendiagramme Ternäre Systeme Folie 25
26 Ternary Four-Phase Equilibrium Class I Isopleths Section at constant ratio x A /x C Section at constant x C VO Phasendiagramme Ternäre Systeme Folie 26
27 Rule of Landau and Palatnik r 1 = r d d + r = dimension of phase diagram r 1 = dimension of boundary between two neighboring phase fields d = number of phases that disappear d + = number of phases that newly appear Section at ~ 50 at% C r = VO Phasendiagramme Ternäre Systeme Folie 27
28 Ternary Four-Phase Equilibrium Class I Solidification eutectic troughs Four-phase plane Alloy X: primary crystallization of single-phase β, secondary crystallization of two-phase mixture α+β, tertiary crystallization of three-phase mixture α+β+γ (most common case) Alloy Y: crystallization of β+γ in equilibrium situation (in non-equilibrium situation probably some liquid would be left over, leading to a secondary crystallization of three-phase α+β+γ as indicated by blue arrows) Alloy Z: primary crystallization of single-phase γ, secondary crystallization of three-phase α+β+γ (only by incidence possible) VO Phasendiagramme Ternäre Systeme Folie 28
29 Ternary Four-Phase Equilibrium Class I Ternary Eutectoid γ α + β + δ VO Phasendiagramme Ternäre Systeme Folie 29
30 Ternary Four-Phase Equilibrium Participation of a binary monotectic Class I L I α + β + L II Isothermal Sections VO Phasendiagramme Ternäre Systeme Folie 30
31 Ternary Four-Phase Equilibrium Class II L + α β + γ A p 1 B II (U) Schematic Liquidus Projection e 2 e 1 C VO Phasendiagramme Ternäre Systeme Folie 31
32 Ternary Four-Phase Equilibrium Class II VO Phasendiagramme Ternäre Systeme Folie 32
33 Ternary Four-Phase Equilibrium Class II Isothermal Sections VO Phasendiagramme Ternäre Systeme Folie 33
34 Ternary Four-Phase Equilibrium T 3 β Class II α α L L L + α β + γ T 4 γ β α L T 5 γ β β α L γ γ VO Phasendiagramme Ternäre Systeme Folie 34
35 Ternary Four-Phase Equilibrium Class II C Isopleths C VO Phasendiagramme Ternäre Systeme Folie 35
36 Ternary Four-Phase Equilibrium Class III L + α + β γ A e 1 B Schematic Liquidus Projection p 2 III (P) p 1 C VO Phasendiagramme Ternäre Systeme Folie 36
37 Ternary Four-Phase Equilibrium Class III L + α + β γ VO Phasendiagramme Ternäre Systeme Folie 37
38 L + α + β γ Ternary Four-Phase Equilibrium Class III Isothermal Sections 1: L+ α + β α + β + γ 2: L+ α + β L + α + γ 3: L+ α + β L + β + γ VO Phasendiagramme Ternäre Systeme Folie 38
39 Ternary Four-Phase Equilibrium Class III Isopleths VO Phasendiagramme Ternäre Systeme Folie 39
40 Congruent Transformations in Ternary Systems Pseudobinary (quasibinary) system VO Phasendiagramme Ternäre Systeme Folie 40
41 Congruent Transformations in Ternary Systems Isothermal Sections Pseudobinary (quasibinary) system VO Phasendiagramme Ternäre Systeme Folie 41
42 Congruent Transformations in Ternary Systems This is not a pseudobinary (quasibinary) system! VO Phasendiagramme Ternäre Systeme Folie 42
43 Congruent Transformations in Ternary Systems Liquidus Projections This is a pseudobinary (quasibinary) system This is not a pseudobinary (quasibinary) system VO Phasendiagramme Ternäre Systeme Folie 43
44 Congruent Transformations in Ternary Systems Congruent transformations can be: pure components (trivial) binary phases with congruent melting point ternary phases with congruent melting point Congruent transformations can give rise to quasibinary (pseudobinary) sections which divide the ternary phase diagram into different parts. For example, the existence of a congruently melting ternary phase (δ) can(!) possibly divide the ternary system into three different, more or less independent, systems if it forms quasibinary sections with the pure components (but this is not necessarily so) VO Phasendiagramme Ternäre Systeme Folie 44
45 Quasibinary Systems Number of possible quasibinary systems: n = b + 2 t + 1 n = No. of independent ternary systems b = No. of binary congruently melting phases t = No. of ternary congruently melting phases n = = 6 Clear Cross Principle VO Phasendiagramme Ternäre Systeme Folie 45
46 Complex Ternary Systems consist of: one-phase equilibria: two-phase equilibria: spaces of any shape spaces having two conjugate surfaces (joined by tie lines at every point) VO Phasendiagramme Ternäre Systeme Folie 46
47 Complex Ternary Systems consist of: three-phase equilibria: a trio of conjugate curves (connected by tie-triangles everywhere) four-phase equilibria: three different types Class I Class II Class III VO Phasendiagramme Ternäre Systeme Folie 47
48 Complex Ternary Systems Ternary Four-Phase Equilibria Class I Class II Class III VO Phasendiagramme Ternäre Systeme Folie 48
49 Complex Ternary Systems Complex Hypothetical System I: δ L + β + γ I II: L + β γ + η I: L α + γ + η I II VO Phasendiagramme Ternäre Systeme Folie 49
50 Complex Ternary Systems Scheil Diagram (Reaction Scheme after Scheil) Ternary A α e 1 I (E) β B e 2 γ e 3 C Ternary A α II (U) p 1 β B γ e 1 p 2 C VO Phasendiagramme Ternäre Systeme Folie 50
51 Complex Ternary Systems Complex Hypothetical System VO Phasendiagramme Ternäre Systeme Folie 51
52 Complex Ternary Systems Example: Ag-Bi-Tl From: Handbook of Ternary Alloy Phase Diagrams, P. Villars, A. Prince, H. Okamoto, Eds., ASM International VO Phasendiagramme Ternäre Systeme Folie 52
53 Complex Ternary Systems Example: In-Ni-Sb From: K.W. Richter, K. Micke, and H. Ipser: Contact Materials for III-V Semiconductors: Phase Equilibria of InSb in the Ternary System In-Ni-Sb ; Mater. Sci. Eng. B 55 (1998), VO Phasendiagramme Ternäre Systeme Folie 53
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