1. PHASE DIAGRAMS  Phase Rule  Types of Phase diagrams  Lever Rule Alloy Phase Equilibria A. Prince Elsevier Publishing Company, Amsterdam (1966) Advanced Reading EQUILIBRIUM PHASE DIAGRAMS

2.  Phase diagrams are an important tool in the armory of an materials scientist  In the simplest sense a phase diagram demarcates regions of existence of various phases. This is similar to a map which demarcates regions based on political, geographical, ecological etc. criteria.  Phase diagrams are maps*  Thorough un derstanding of phase diagrams is a must for all materials scientists  Phase diagrams are also referred to as “EQUILIBRIUM PHASE DIAGRAMS”  This usage requires special attention: though the term used is “Equilibrium”, in practical terms the equilibrium is NOT GLOBAL EQUILIBRIUM but MICROSTRUCTURAL LEVEL EQUILIBRIUM (explanation of the same will be considered later)  This implies that any microstructural information overlaid on a phase diagram is for convenience and not implied by the phase diagram.  The fact that Phase Diagrams represent Microstructural Level equilibrium is often not stressed upon. Phase Diagrams * there are many other maps that a material scientist will encounter like creep mechanism maps, various kinds of materials selection maps etc.

3.  Broadly two kinds of phase diagrams can be differentiated* → those involving time and those which do not involve time (special care must be taken in understanding the former class- those involving time).  In this chapter we shall deal with the phase diagrams not involving time. This type can be further sub-classified into:  Those with composition as a variable (e.g. T vs %Cu)  Those without composition as a variable (e.g. P vs T)  Temperature-Composition diagrams (i.e. axes are T and composition) are extensively used in materials science and will be considered in detail in this chapter. Also, we shall restrict ourselves to structural phases (i.e. phases not defined in terms of a physical property)**  Time-Temperature-Transformations (TTT) diagrams and Continuous-Cooling- Transformation (CCT) diagrams involve time. These diagrams are usually designed to have an overlay of Microstructural information (including microstructural evolution). These diagrams will be considered in the chapter on Phase Transformations. * this is from a convenience in understanding point of view ** we have seen before that phases can be defined based either on a geometrical entity or a physical property (sometimes phases based on a physical property are overlaid on a structural phase diagram- e.g. in a Fe-cementite phase diagram ferromagnetic phase and curie temperatures are overlaid)

4. AtomStructure Crystal Electro- magnetic MicrostructureComponent Thermo-mechanical Treatments PhasesDefects+ Casting Metal Forming Welding Powder Processing Machining Vacancies Dislocations Twins Stacking Faults Grain Boundaries Voids Cracks + Residual Stress Processing determines shape and microstructure of a component & their distributions Since terms like Phase and Microstructure are key to understanding phase diagrams, let us have a re-look at the figure we considered before Click here to know more about microstructures

5. DEFINITIONS Components of a system Independent chemical species which comprise the system: These could be: Elements, Ions, Compounds E.g.  Au-Cu system : Components → Au, Cu (elements)  Ice-water system : Component → H 2 O (compound)  Al 2 O 3 – Cr 2 O 3 system : Components → Al 2 O 3, Cr 2 O 3 Let us start with some basic definitions: This is important to note that components need not be just elements!! Note that components need not be only elements

6. Phase Physically distinct, chemically homogenous and mechanically separable region of a system (e.g. gas, crystal, amorphous...).  Gases Gaseous state always a single phase → mixed at atomic or molecular level  Liquids ►Liquid solution is a single phase → e.g. NaCl in H 2 O ► Liquid mixture consists of two or more phases → e.g. Oil in water (no mixing at the atomic/molecular level)  Solids  In general due to several compositions and crystals structures many phases are possible  For the same composition different crystal structures represent different phases. E.g. Fe (BCC) and Fe (FCC) are different phases  For the same crystal structure different compositions represent different phases. E.g. in Au-Cu alloy 70%Au-30%Cu & 30%Au-70%Cu are different phases This is the typical textbook definition which one would see!! Three immiscible liquids

7. What kinds of Phases exist?  Based on state  Gas, Liquid, Solid  Based on atomic order  Amorphous, Quasicrystalline, Crystalline  Based on Band structure  Insulating, Semi-conducting, Semi-metallic, Metallic  Based on Property  Paraelectric, Ferromagnetic, Superconducting, …..  Based on Stability  Stable, Metastable, (also- Neutral, unstable)  Also sometimes- Based on Size/geometry of an entity  Nanocrystalline, mesoporous, layered, …  We have already seen the ‘official’ definition of a phase: Physically distinct, chemically homogenous and mechanically separable region of a system.  However, the term phase is used in diverse contexts and we list below some of these.

8. Microstructure Structures requiring magnifications in the region of 100 to 1000 times OR The distribution of phases and defects in a material Grain The single crystalline part of polycrystalline metal separated by similar entities by a grain boundary Phase Transformation is the change of one phase into another. E.g.:► Water → Ice ►  - Fe (BCC) →  - Fe (FCC)   - Fe (FCC) →  - Fe (ferrite) + Cementite (this involves change in composition) ► Ferromagnetic phase → Paramagnetic phase (based on a property) Phase transformation Again this is a typical textbook definition which has been included for…!! An alternate definition based on magnification  (Phases + defects + residual stress) & their distributions

9. Phase diagram Map demarcating regions of stability of various phases. or Map that gives relationship between phases in equilibrium in a system as a function of T, P and composition (the restricted form of the definition sometime considered in materials textbooks) Variables / Axis of phase diagrams  The axes can be:  Thermodynamic (T, P, V), Other possibilities include magnetic field intensity (H), electric field (E) etc.  Kinetic (t) or  Composition variables (C, %x) (composition is usually measure in weight%, atom% or mole fraction)  In single component systems (unary systems) the usual variables are T & P  In phase diagrams used in materials science the usual variables are:T & %x  In the study of phase transformation kinetics Time Temperature Transformation (TTT) diagrams or Continuous Cooling Transformation (CCT) diagrams are also used where the axis are T & t

10.  Phase diagrams are also called Equilibrium Phase Diagrams.  Though not explicitly stated the word ‘Equilibrium’ in this context usually means Microstructural level equilibrium and NOT Global Equilibrium.  Microstructural level equilibrium implies that microstructures are ‘allowed to exist’ and the system is not in the global energy minimum state.  This statement also implies that:  Micro-constituents* can be included in phase diagrams  Certain phases (like cementite in the Fe-C system) maybe included in phase diagrams, which are not strictly equilibrium phases (cementite will decompose to graphite and ferrite given sufficient thermal activation and time)  Various defects are ‘tolerated’ in the product obtained. These include defects like dislocations, excess vacancies, internal interfaces (interphase boundaries, grain boundaries) etc.  Often cooling ‘lines/paths’ are overlaid on phase diagrams- strictly speaking this is not allowed. When this is done, it is implied that the cooling rate is ‘very slow’ and the system is in ‘~equilibrium’ during the entire process. (Sometimes, even fast cooling paths are also overlaid on phase diagrams!) Important points about phase diagrams (Revision + extra points) * will be defined later

11. The GIBBS PHASE RULE F = C  P + 2 For a system in equilibrium F  C + P = 2 or F – Degrees of Freedom C – Number of Components P – Number of Phases The Phase rule is best understood by considering examples from actual phase diagrams as shown in some of the coming slides Degrees of Freedom: A general definition  In response to a stimulus the ways in which the system can respond corresponds to the degrees of freedom of the system  The phase rule connects the Degrees of Freedom, the number of Components in a system and the number of Phases present in a system via a simple equation.  To understand the phase rule one must understand the variables in the system along with the degrees of freedom.  We start with a general definition of the phrase: “degrees of freedom” The phase rule

12. Variables in a Phase Diagram  C – No. of Components  P – No. of Phases  F – No. of degrees of Freedom  Variables in the system = Composition variables + Thermodynamic variables  Composition of a phase specified by (C – 1) variables (e.g. If the composition is expressed in %ages then the total is 100%  there is one equation connecting the composition variables and we need to specify only (C  1) composition variables)  No. of variables required to specify the composition of all Phases: P(C – 1) (as there are P phases and each phase needs the specification of (C  1) variables)  Thermodynamic variables = P + T (usually considered) = 2 (at constant Pressure (e.g. atmospheric pressure) the thermodynamic variable becomes 1)  Total no. of variables in the system = P(C – 1) + 2  F < no. of variables  F < P(C – 1) + 2

13. It is worthwhile to clarify a few terms at this stage:  Components ‘can’ go on to make a phase (ofcourse one can have single component phases as well- e.g. BCC iron phase, ferromagnetic iron phase etc.)  Phases ‘can’ go on to make a microconstituent  Microconstituents ‘can’ go on to make a microstructure (ofcourse phases can also directly go on to make a microstructure)  For a system in equilibrium the chemical potential of each species is same in all the phases.  If , , ,… are phases, then:  A (  ) =  A (  ) =  A (  )….  Suppose there are 2 phases (  &  phases) and 3 components (A,B,C) in each phase. Then:  A (  ) =  A (  ),  B (  ) =  B (  ),  C (  ) =  C (  )  i.e there are 3 equations For each component there are (P  1) equations and for C components the total number of equations is C(P  1) In the above example the number of equations is 3(2  1)=3 equations.  F = (Total number of variables)  (number of relations between variables) = [P(C – 1) + 2]  [C(P  1)] = C  P + 2  In a single phase system F = no. of variables.  P ↑  F ↓ (for a system with fixed number of components as the number phases increases the degrees of freedom decreases) The Gibbs Phase Rule F = C  P + 2

14. Degrees of Freedom = What you can controlWhat the system controls  F = C + 2P  Can control the no. of components added and P & T System decided how many phases to produce given the conditions A way of understanding the Gibbs Phase Rule: The degrees of freedom can be thought of as the difference between what you (can) control and what the system controls F = C  P + 2

15. No. of phases Total variables P(C – 1) +2 Degrees of Freedom C – P +2 Degrees of Freedom C – P +1 1332 2421 3510 460Not possible C = 2 No. of phases Total variables P(C – 1) +2 Degrees of Freedom C – P +2 Degrees of Freedom C – P +1 1443 2632 3821 41010 C = 3 3 components 2 components Variation of the number of degrees of freedom with number of components and number of phases Phase rule with pressure fixed (at say 1 atm) 3 phase co- existence is an invariant point

16. The Gibbs phase rule and its geometrical cousin  There is an interesting connection between the Gibbs Phase Rule and the Euler Rule for Convex Polyhedra.  The Euler’s formula for convex polyhedra can be considered as the geometrically equivalent Gibbs Phase rule. The diagram below brings out the analogous variables. V 0  V 1 + V 2 = 2 V  E + F = 2 or V 0 = V → Vertices V 1 = E → Edges V 2 = F → Faces Edges ~ Components Vertices ~ degrees of Freedom Faces ~ Phases F  C + P = 2 Gibbs Phase Rule 8 – 12 + 6 = 2 For the cube V  E + F = 2

17. Single component phase diagrams (Unary)  Let us start with the simplest system possible: the unary system wherein there is just one component.  Though there are many possibilities even in unary phase diagrams (in terms of the axis and phases), we shall only consider a T-P unary phase diagram.  Let us consider the Fe unary phase diagram as an illustrative example.  Apart from the liquid and gaseous phases many solid phases are possible based on crystal structure. (Diagram on next page).  Note that the units of x-axis are in GPa (i.e. high pressures are needed in the solid state and liquid state to see any changes to stability regions of the phases).  The Gibbs phase rule here is: F = C – P + 2. (2 is for T & P).  Note that how the phase fields of the open structure (BCC- one in the low T regime (  ) and one in the high T regime (  )) diminish at higher pressures. In fact  - phase field completely vanishes at high pressures.  The variables in the phase diagram are: T & P (no composition variables here!).  Along the 2 phase co-existence lines the DOF (F) is 1 → i.e. we can chose either T or P and the other will be automatically fixed.  The 3 phase co-existence points are invariant points with F = 0. (Invariant point implies they are fixed for a given system).

18. 1535 1410  (BCC)  (HCP)  (FCC)  (BCC) Liquid Gas Pressure (GPa) → Temperature (ºC) → Triple points: 3 phase coexistence F = 1 – 3 + 2 = 0  triple points are fixed points of a phase diagram (we cannot chose T or P) Two phase coexistence lines F = 1 – 2 + 2 = 1  we have only one independent variable (we can chose one of the two variables (T or P) and the other is ‘automatically’ fixed by the phase diagram) Single phase regions F = 1 – 1 + 2 = 2  T and P can both be varied while still being in the single phase region F = C – P + 2 The maximum number of phases which can coexist in a unary P-T phase diagram is 3 Note the P is in GPa “Very High pressures are required for things to happen in the solid state”

19. Understanding aspects of the iron unary phase diagram  The degrees of freedom for regions, lines and points in the figure are marked in the diagram shown before  The effect of P on the phase stability of various phases is discussed in the diagram below  It also becomes clear that when we say iron is BCC at RT, we mean at atmospheric pressure (as evident from the diagram at higher pressures iron can become HCP)  (BCC)  (HCP)  (FCC)  (BCC) Liquid Gas Pressure (GPa) → Temperature (ºC) → This line slopes upward as at constant T if we increase the P the gas will liquefy as liquid has lower volume (similarly the reader should draw horizontal lines to understand the effect of pressure on the stability of various phases- and rationalize the same). These lines slope downward as: Under higher pressure the phase with higher packing fraction (lower volume) is preferred Increase P and gas will liquefy on crossing phase boundary Phase fields of non-close packed structures shrink under higher pressure Phase fields of close packed structures expand under higher pressure Usually (P = 1 atm) the high temperature phase is the loose packed structure and the RT structure is close packed. How come we find BCC phase at RT in iron?

20. Binary Phase Diagrams  Binary implies that there are two components.  Pressure changes often have little effect on the equilibrium of solid phases (unless ofcourse we apply ‘huge’ pressures).  Hence, binary phase diagrams are usually drawn at 1 atmosphere pressure.  The Gibbs phase rule is reduced to:  Variables are reduced to: F = C – P + 1. (1 is for T).  T & Composition (these are the usual variables in Materials Phase Diagrams) F = C  P + 1 Phase rule for condensed phases For T  In the next page we consider the possible binary phase diagrams. These have been classified based on:  Complete Solubility in both liquid & solid states  Complete Solubility in both liquid state, but limited solubility in the solid state  Limited Solubility in both liquid & solid states.

21. Complete Solubility in both liquid state, but limited solubility in the solid state Overview of Possible Binary Phase diagrams Isomorphous Isomorphous with phase separation Isomorphous with ordering Complete Solubility in both liquid & solid states Limited Solubility in both liquid & solid states Eutectic Peritectic Liquid State Solid State analogue Eutectoid Peritectoid Monotectic Syntectic Monotectoid Solid state analogue of Isomorphous

22.  We have already seen that the reduced phase rule at 1Atm pressure is: F = C – P + 1.  The ‘one’ on RHS above is T.  The other two variables are:  Composition of the liquid (C L ) and composition (C S ) of the solid. In a fully solid state reaction:  Composition of one solid (C S1 ) and composition of the other solid (C S2 ).  The compositions are defined with respect to one of the components (say B): C L B, C S B  The Degrees of Freedom (DOF, F) are defined with respect to these variables. What are the variables/DOF in a binary phase diagram?

23. A B System with complete solid & liquid solubility: ISOMORPHOUS SYSTEM %B → T → M.P. of B M.P. of A C = 1 P = 2 F = 0 C = 2 P = 2 F = 1 A B The two component region expands with one degree of freedom i.e. we can chose one variable – say the T F = C – P + 1  Let us start with an isomorphous system with complete liquid and solid solubility (as we have noted complete solid solubility is a little difficult to get– Hume-Rothery rules have to be satisfied).  If we try to draw a straight line connecting the melting point of A and B (the two components)- we note that the degree of freedom for any intermediate composition is 1  this line expands into a region (which is a two phase mixture of liquid and solid).  Pure components melt at a single temperature, while alloys in the isomorphous system melt over a range of temperatures*. I.e. for a given composition solid and liquid will coexist over a range of temperatures when heated. * Assuming that the components do not decompose or sublime.  melting points are fixed!

24. Model Isomorphous phase diagram We mention some important points here (may be/have been reiterated elsewhere!):  Such a phase diagram forms when there is complete solid and liquid solubility.  The solid mentioned is crystalline.  The solid + liquid region is not a semi-solid (like partly molten wax or silicate glass). It is a crystal of well defined composition in equilibrium with a liquid of well defined composition.  Both the solid and the liquid and the solid (except pure A and pure B) have both A and B components in them.  A and B components could be pure elements (like in the Ag-Au, Au-Pd, Au-Ni, Ge-Si) or compounds (like Al 2 O 3 -Cr 2 O 3 ).  At low temperatures the picture may not be ‘ideal’ as presented in the diagram below and we may have phase separation (Au-Ni system) or have compound formation (for some compositions) (Au-Pd system). These cases will be considered later.  Each solid, with a different composition is a different phase. The area marked solid in the phase diagram is a phase field.  If heated further the liquid will vaporize, this part of the phase diagram is usually not shown in the diagrams considered. Note that between two single phase regions there is a two phase region (for the alloy) (except for special cases)

25. A B %B → T → M.P. of B M.P. of A C = 1 P = 2 F = 0 C = 2 P = 1 F = 2 C = 2 P = 1 (liquid) F = 2 Variables → T, C S B  2 Variables → T, C L B  2 Variables → T, C L B, C S B  3 F = 2 – P F= 3 – P C = 2 P = 2 F = 1 F = 2 – P F = C – P + 1 Now let us map the variables and degrees of freedom in varions regions of the isomorphous phase diagram  T and Composition can both be varied while still being in the single phase region  in the two phase region, if we fix T (and hence exhaust our DOF), the composition of liquid and solid in equilibrium are automatically fixed (i.e. we have no choice over them).  Alternately we can use our DOF to chose C L → then T and C S are automatically fixed. For pure components at any T For alloys Solid Solid + Liquid Liquid Disordered (substitutional) solid solutions For pure components all transformation temperatures (BCC to FCC, etc.) are fixed (i.e. zero ‘F’)

26. Gibbs free energy vs composition plot at various temperatures: Isomorphous system  As we know at constant T and P the Gibbs free energy determines the stability of a phase. Hence, a phase diagram can be constructed from G-composition (  G mixing -C) curves at various temperatures. For an isomorphous system we need to chose 5 sample temperatures: (i) T 1 > T A, (ii) T 2 =T A, (iii) T A >T 3 >T B, (iv) T 4 =T B, (v) T 5 <T B. G of L lower than  for all compositions and hence L is stable G of L lower than  for all compositions except for pure A. For compositions between X 1 and X 2 the common tangent construction gives the free energy of the L+  mixture 5 How to get  G versus composition curves→ Click here to know more. How to get  G versus composition curves→ Click here to know more.

27. Al 2 O 3 Cr 2 O 3 %Cr 2 O 3 → T (ºC) → 2000 2100 2200 1030 507090 L L + S Solidus Liquidus S Isomorphous Phase Diagram: an example  A and B must satisfy Hume-Rothery rules for the formation of ‘extended’ solid solution.  Examples of systems forming isomorphous systems: Cu-Ni, Ag-Au, Ge-Si, Al 2 O 3 -Cr 2 O 3.  Note the liquidus (the line separting L & L+S regions) and solidus (the line separating L+S and S regions) lines in the figure. Schematics Note that the components in this case are compounds

28. ISOMORPHOUS PHASE DIG. Points to be noted:  Pure components (A,B) melt at a single temperature. (General) Alloys melt over a range of temperatures (we will see some special cases soon).  Isomorphous phase diagrams form when there is complete solid and liquid solubility.  Complete solid solubility implies that the crystal structure of the two components have to be same and Hume-Rothery rules have to be followed.  In some systems (e.g. Au-Ni system) there might be phase separation in the solid state (i.e. the complete solid solubility criterion may not be followed) → these will be considered later in this chapter as a variation of the isomorphous system (with complete solubility in the solid and the liquid state).  Both the liquid and solid contain the components A and B.  In Binary phase diagrams between two single phase regions there will be a two phase region → In the isomorphous diagram between the liquid and solid state there is the (Liquid + Solid) state.  The Liquid + Solid state is NOT a ‘semi-solid’ state → it is a solid of fixed composition and structure, in equilibrium with a liquid of fixed composition.  In the single phase region the composition of the alloy is ‘the composition’. In the two phase region the composition of the two phases is different and is NOT the nominal composition of the alloy (but, is given by the lever rule). Lever rule is considered next. HUME ROTHERY RULES Click here to know more about

29.  Say the composition C 0 is cooled slowly (equilibrium)  At T 0 there is L + S equilibrium  Solid (crystal) of composition C 1 coexists with liquid of composition C 2 Tie line and Lever Rule Given a temperature and composition- how do we find the fraction of the phases present along with the composition?  We draw a horizontal line (called the Tie Line) at the temperature of interest (say T 0 ).  Tie line is XY.  Note that tie lines can be drawn only in the two phase coexistence regions (fields). Though they may be extended to mark the temperature.  To find the fractions of solid and liquid we use the lever rule.

30. %B → A B T → L L + S S C0C0 C1C1 C2C2 T0T0 Cooling Fulcrum of the lever Arm of the lever proportional to the solid Arm of the lever proportional to the liquid Note: strictly speaking cooling curves cannot be overlaid on phase diagrams Tie line At T 0  The fraction of liquid (f l ) is  (C 0  C 1 )  The fraction of solid (f s ) is  (C 2  C 0 )  We draw a horizontal line (called the Tie Line) at the temperature of interest (say T 0 ).  The portion of the horizontal line in the two phase region is akin to a ‘lever’ with the fulcrum at the nominal composition (C 0 ).  The opposite arms of the lever are proportional to the fraction of the solid and liquid phases present (this is the lever rule). Note that tie line is drawn within the two phase region and is horizontal.

31. Expanded version Extended tie line C0C0 C1C1 C2C2 T0T0 Fulcrum of the lever Arm of the lever proportional to the solid Arm of the lever proportional to the liquid At T 0  The fraction of liquid (f l ) is proportional to (C 0  C 1 ) → AC  The fraction of solid (f s ) is proportional to (C 2  C 0 ) → CB B A C

32. How does the fraction (or %) of solid and liquid change as we cool (in the two phase region)? As we cool from T 1 to T 9 in the two phase region:  The first solid appears at T 1 (effectively the fraction of solid at T 1 is zero).  The fraction of solid increases according the solid line in the plot below and correspondingly the fraction of liquid decreases.  The last bit of liquid solidifies at T 9.  The composition of the solid and liquid in equilibrium with it keeps on changing as we cool (as in upcoming slide).  The total percentage (L + S) is always 100%.

33. For a composition C 0  At T 0 → Both the liquid and the solid phases contain both the components A and B  To reiterate: The state is NOT semi-solid but a mixture of a solid of a definite composition (C 1 ) with a liquid of definite composition (C 2 )  If the alloy is slowly cooled (maintaining ~equilibrium) then in the two phase region (liquid + solid region) the composition of the solid will move along the brown line and the composition of the liquid will move along the blue line.  The composition of the solid and liquid are changing as we cool! Points to be noted

34. Isomorphous Phase Diagrams Note here that there is solid solubility, but it is not complete at low temperatures (below the peak of the  1 +  2 phase field dome) (we will have to say more about that soon) Note that  Ag & Au are so ‘similar’ that the phase diagram becomes a thin lens (i.e. any alloy of Au & Ag melts over a small range of temperatures– as if it were ‘nearly’ a pure metal!!).  Any composition melts above the linear interpolated melting point. T1T1 Below T 1 (820  C) for some range of compositions the solid solubility of Au in Ni (and vice-versa) is limited. Any composition melts above the linearly interpolated melting point

35. Funda Check One phase or many phases? → the concept of the phase field  We have noted that different crystal structures of the same component are different phases.  Different compositions of the same crystal structure differ in lattice parameter and constitute different phases.  Sometimes in the region of stability of a ‘phase’ (say  ), different compositions are referred to (‘casually’) as  -phase.  The region should be called a Phase Field and different compositions are actually different phases.  The solid solution based on a component is often written in brackets. E.g. (Cu) marked in the phase diagram implies a solid solution of Cu with another component (say Ag).

36. Funda Check Is the lever rule right? (I.e. does it give the right phase fractions?) Let A and B be the two components constituting the binary phase diagram. Consider co-existing phases  (with composition C 1 ) and  (with composition C 2 ) with a mean composition C 0. ‘f’ denotes the fraction of a particular phase. Let be the compositions expressed in terms of the percentage of B.

37. Extensions of the simple isomorphous system: Congruently melting alloys  Congruently melting alloys  just like a pure metal  Is the DOF 1? No: in requiring that C L B = C S B we have exhausted the degree of freedom. Hence T is automatically fixed → DOF is actually zero! Tie line has shrunk to a point! Variables → T, C L B, C S B  3 C = 2 P = 2 F = 1?? (see below)  We have seen that a pure metal melts at a single temperature (Why?!!).  An alloy typically melts over a range of temperatures. However, there are special compositions which can melt at a single temperature like a pure metal. One of these is the congruent melting composition- in a variation of the isomorphous phase diagram. Some systems show this type of behaviour.  Intermediate compounds also have this feature as we shall see later. Elevation in MP Depression in MP Case A Case B

38.  These isomorphous phase diagrams with congruent melting compositions can be understood as two simple isomorphous diagrams with C 0 as one of the components: i.e. A-C 0 is one and C 0 -B is the other. %C 0 →

39.  A  A and B  B bonds stronger than A  B bonds  Liquid stabilized → Phase separation in the solid state  A  B bonds stronger than A  A and B  B bonds  Solid stabilized → Ordered solid formation E.g. Au-Ni Extensions of the simple isomorphous system: What does this imply w.r.t the solid state phases?  Elevation in the MP means that the solid state is ‘more stable’ (crudely speaking the ‘ordered state is more stable’) → ordering reaction is seen at low T.  Depression in MP ‘means’ the liquid state (disordered) is more stable → phase separation is seen at low T. (Phase separation can be ‘thought of’ as the ‘opposite’ of ordering. Ordering (compound formation) occurs for  ve values of  H mix, while phase separation is favoured by +ve values of  H mix. Case A Case B

40. Examples of isomorphous systems with phase separation and compound formation Au-Pd system with 3 compounds Au-Pt system with phase separation at low temperatures Au-Ni: model system to understand phase separation Phase separation in a AlCrFeNi alloy (with composition Al 28.5 Cr 27.3 Fe 24.9 Ni 19.3 ) into two BCC phases

41. Ti-Zr: With solid state analogue of isomorphous phase diagram Congruent melting composition Congruent transformation between HCP & BCC phases

42. Congruent transformations We have seen two congruent transformations (transformations which occur without change in composition). The list is as below.  Melting point minimum  Melting point maximum  Order – disorder transformation  Formation of an intermediate phase Melting point maximum Order disorder transformation Formation of an intermediate phase

43.  Very few systems exhibit an isomorphous phase diagram (usually the solid solubility of one component in another is limited).  Often the solid solubility is severely limited- though the solid solubility is never zero (due to entropic reasons).  In a simple eutectic system (binary), there is one composition at which the liquid freezes to two solids at a single temperature. This is in some sense similar to a pure solid which freezes at a single temperature (unlike a pure substance the freezing produces a two solid phases- both of which contain both the components).  The term EUTECTIC means Easy Melting → The alloy of eutectic composition freezes at a lower temperature than the melting points of the constituent components.  This has important implications→ e.g. the Pb-Sn * eutectic alloy melts at 183  C, which is lower than the melting points of both Pb (327  C) and Sn (232  C)  can be used for soldering purposes (as we want to input least amount of heat to solder two materials).  In the next page we consider the Pb-Sn eutectic phase diagram.  As noted before the components need not be only elements. E.g. in the A-Cu system a eutectic reaction is seen between  (Solid solution of Cu in Al) and  (Al 2 Cu- a compound). Eutectic Phase Diagram * Actually  -  eutectic alloy (or (Pb)-(Sn) eutectic alloy)

44. PbSn %Sn → T (ºC) → 100 200 300 1030 507090 L   +   L +   + L Liquidus Solvus Solidus 18% 62% 97% 183  C 232  C 327  C Eutectic reaction (the proper way of writing the reaction) Eutectic reaction L →  +  C eutectic = C E T eutectic = T E E F D Note that Pb is CCP, while Sn at RT is Tetragonal (tI4, I4 1 amd) → therefore complete solid solubility across compositions is ruled out!!

45. Note the following points:   and  are terminal solid solutions (usually terminal solid solutions are given symbols (  and  )) ; i.e.  is a solid solution of B (Sn) in A (Pb). (In some systems the terminal solid solubility may be very limited: e.g. the Bi-Cd system).   has the same crystal structure as that of A (Pb in the example below) and  has the same crystal structure as B (Sn in the example below).  Typically, in eutectic systems the solid solubility increases with temperature till the eutectic point (i.e. we have a ‘sloping solvus line’). In many situations the solubility of component B in A (and vice- versa) may be very small.  The Liquidus, Solidus and Solvus lines are as marked in the figure below.

46. AB %B → T (ºC) 100 200 300 1030 507090 L   +   L +   + L Eutectic reaction L →  +  Increasing solubility of B in A with ↑T C = 2 P = 3 F = 0  At the eutectic point E (fig. below) → 3 phases co-exist: L,  &  The number of components in a binary phase diagram is 2  the number of degrees of freedom F = 0.  This implies that the Eutectic point is an Invariant Point  for a given system it occurs at a fixed composition and temperature.  For a binary system the line DF is a horizontal line.  Any composition lying between D and F will show eutectic solidification at least in part (for composition E the whole liquid will solidify by the eutectic reaction as shown later).  The percentage of  and  produced by eutectic solidification at E is found by considering DF* as a lever with fulcrum at E. E F D E F D * Actually just below DF as tie lines are drawn in a two phase region

47. Further points:  Extension of the boundary line between a single phase region and adjacent two-phase region, should lie in the two phase region (red dashed lines as in figure below).  The eutectic reaction is a phase reaction and not a chemical reaction.

48. Examples of Eutectic microstructures As pointed out before microstructural information is often overlaid on phase diagrams. These represent microstructures which evolve on slow cooling. Al-Al 2 Cu lamellar eutectic Composition plot across lamellae Pb-Sn lamellar eutectic Though we label the microstructure as Pb-Sn lamellar eutectic it is actually a  -  eutectic. Al 2 Cu (note that one of the components is a compound!) (Al)

49. Gibbs free energy vs composition plot at various temperatures: Eutectic system

50. A B L L +   A B L  11 22  1 +  2 A B L 11 22 L +   1 +  2 A B L 11 22 L +  1  1 +  2 L +  2 Isomorphous to Eutectic  A  A and B  B bonds stronger than A  B bonds  Liquid stabilized A eutectic system can be visualized as arising from an isomorphous system with depression in MP, as below.

51.  To reiterate an important point: phase diagrams do not contain microstructural information (i.e. they cannot tell you what is the microstructure produced by cooling). Often microstructural information is overlaid on phase diagrams for convenience. Hence, strictly cooling is not in the domain of phase diagrams- but we can overlay such information keeping in view the assumptions involved.  For the following explanations refer to the Pb-Sn eutectic diagram and the diagram considered next. Solidification of three range(s) of compositions need to be understood: (i) to the left of C E , (ii) between C E  and C E and (iii) C E.  (iii)  On cooling an eutectic composition (C E or C 3E in the next page), at the eutectic temperature (T E ), both the constituents (say  &  ) of the eutectic reaction will ‘simultaneously’ form from the liquid. In practice (based on other factors) one of the constituents may form before the other. The entire solidification takes place at a single temperature- T E.  On slow cooling, the microstructure produced by the eutectic reaction could be lamellar, ‘Chinese script’ (like), etc. This distribution of phases (say lamellar) is called a microconstituent. Solidification of Eutectic and Off-eutectic compositions

52.  (i)  On cooling a composition to the left of C E  →  The  phase will start solidifying from the liquid and between 1a and 1b there will be a mixture of  and L.  Between 1b and 1c there will be only  phase.  Below 1c (as the solubility of B in  reduces)  phase will precipitate out of  (and we will have a mixture of  +  ).  (ii)  On cooling a composition between C E  and C E →  Between 2a and 2b, the  phase will start solidifying from the liquid and we will have a mixture of L +  (this  is known as the proeutectic-   implying that it is ‘pre’-eutectic  ). On cooling towards T E the composition of the liquid travels along the 2a-E curve.  On reaching T E (i.e. point 2b) the liquid remaining after the solidification of proeutectic-  will solidify as a eutectic mixture (according to the eutectic reaction).  For a composition between C E and C E  the situation is similar to case (ii) above except that we will get proeutectic-  instead of proeutectic- . And for a composition beyond C E  the situation is similar to case (i) above (except that we will get  instead of  ).  The important point to note that even for off-eutectic compositions between C E  and C E  part of the liquid solidifies by the eutectic reaction

53. C1C1 C2C2 C3C3 C4C4 The solidification sequence of C 4 will be similar to C 2 except that the proeutectic phase will be  Pb-Sn eutectic

54. Eutectic system without terminal solid solubility: Bi-Cd Special eutectic diagrams Technically this is incorrect: there has to be some terminal solid solubility  Technically it is incorrect to draw eutectic phase diagrams with zero solid solubility.  This would imply that a pure component (say Bi in the example considered) melts over a range of temperatures (from ‘P’ to 271.4  C) → which is wrong.  Also, let us consider an example of a point ‘P’ (which lies on the ‘eutectic line’ PQ). At ‘P’ the phase rule becomes: F = C–P+1 = 1–3+1 = –1 !!!  Note that the above is an alternate way of arriving at the obvious contradiction that at ‘P’ on one hand we are saying that there is a pure component and on the other hand we are considering a three phase equilibrium (which can happen only for Bi-Cd alloys). P Q ‘Correct’ version of the diagram with some terminal solid solubility.

55. Eutectic system with terminal solid solubility on one side: Ag-Ge Special eutectic diagrams With terminal solid solubility on both sides

56. Solved Example During the solidification of a off eutectic (Pb-Sn) composition (C 0 ), 90 vol.% of the solid consisted of the eutectic mixture and 10vol.% of the proeutectic  phase. What is the value of C 0 ? Density data for  and  :    = 10300 Kg/m 3    = 7300 Kg/m 3 Let us start with some observations:  Pb is ‘heavier’ than Sn and hence the density of  is more than that of .  Since the proeutectic phase is   the composition is hyperpeutectic (towards the Sn side).  The volume fractions (in %) are usually calculated by taking the area fractions by doing metallography (microstructure) and then converting it into volume fractions (usually volume fraction is assumed to be equal to area fractions). Eutectic Data:  183  C  62 wt.% Sn

57. What is meant by ‘microstructural level equilibrium’? Funda Check let us understand the concept using an example considered before. During the eutectic reaction (during slow cooling) a lamellar micro constituent is obtained. This results in a huge amount of interfacial area between the two phases (Al, Al 2 Cu), which will result in a high value of interfacial energy. Fig.1: Al-Al 2 Cu eutectic The equilibrium state would correspond to the schematic as shown below. Since we ‘tolerate’ the microstructure as in Fig.1 (and do not take the system to the global energy minimum state), the equilibrium considered in typical phase diagrams are microstructural level equilibrium. Polyhedral crystals

58. Peritectic Phase Diagram  Like the eutectic system, the peritectic reaction is found in systems with complete liquid solubility but limited solid solubility.  In the peritectic reaction the liquid (L) reacts with one solid (  ) to produce another solid (  ).  L +   .  Since the solid  forms at the interface between the L and the , further reaction is dependent on solid state diffusion. Needless to say this becomes the rate limiting step and hence it is difficult to ‘equilibrate’ Peritectic reactions (as compared to say eutectic reactions). Figure below.  In some peritectic reactions (e.g. the Pt-Ag system- next page), the (pure)  phase is not stable below the peritectic temperature (T P = 1186  C for Pt-Ag system) and splits into a mixture of (  +  ) just below T P.

59. Pt-Ag Peritectic system Peritectic reaction L +  →   Melting points of the components vastly different.  Pt-Ag is perhaps not a good example of a peritectic system– obvious looking at the  phase field (not stable below the peritectic composition).   Note that below T P pure  is not stable and splits into (  +  ) Formal way of writing the peritectic reaction

60. Isomorphous to Peritectic A B L  L +  A B L L +  1 (  2) 11 22  1 +  2 A B L L +  1 11 22  1 +  2 L  L +  11 22  1 +  2

61.  Some phase diagrams are technologically important and via these further aspects of the utility of phase diagrams can be learnt.  The Fe-C (or more precisely the Fe- Fe 3 C) diagram is an important one. Cementite is a metastable phase and ‘strictly speaking’ should not be included in a phase diagram. But the decomposition rate of cementite is small and hence can be thought of as ‘stable enough’ to be included in a phase diagram. Hence, we typically consider the Fe- Fe 3 C part of the Fe-C phase diagram.  Another technologically (and from a perspective of physical Metallurgy) important phase diagram is the Al-Cu system (especially the Al rich end of the phase diagram). Examples: Some important phase diagrams

62.  A portion of the Fe-C diagram- the part from pure Fe to 6.67 wt.% carbon (corresponding to cementite, Fe 3 C)- is technologically very relevant.  Cementite is not a equilibrium phase and would tend to decompose into Fe and graphite. This reaction is sluggish and for practical purpose (at the microstructural level) cementite can be considered to be part of the phase diagram. Cementite forms as it nucleates readily as compared to graphite.  Compositions up to 1.5%C are called steels and beyond 2%C are called cast irons. In reality the classification should be based on ‘castability’ and not just on carbon content.  Heat treatments can be done to alter the properties of the steel by modifying the microstructure → we will learn about this in the chapter on Phase Transformations. This may involve production of metstable phases like martensite (not found in the equilibrium phase diagram).  As before we will use ‘slow’ cooling curves to ‘see’ the microstructures produced. The part of the phase diagram of interest is: (i) with less than ~2% C and (ii) less that ~1100  C. Phases of interest are listed in the table below. The Fe-Cementite phase diagram PhaseStructure Austenite (  ) FCC (CCP) Ferrite (  ) BCC Cementite (Fe 3 C)Orthorhombic Crystal structure of Cementite

63. Wt.% C → T → Fe Fe 3 C 6.67 4.3 0.8 0.16 2.06 Peritectic L +  →  Eutectic L →  + Fe 3 C Eutectoid  →  + Fe 3 C L  L +    + Fe 3 C  1493ºC 1147ºC 723ºC Fe-Cementite diagram 0.025 RT~0.008 Three reactions are seen in the Fe rich side: (i) Peritectic, (ii) Eutectic & (iii) Eutectoid. Of the three reactions the eutectoid reaction is technologically important A1A1 A3A3 A cm Note: we can also draw the true Fe-C diagram (not shown here). Sometime one of these diagrams is shown in dashed lines and overlaid on a single diagram. L  (BCC)  (FCC)  (BCC) 912ºC For Pure Fe 1394ºC 1538ºC

64. The portion of the phase diagram, which is technologically relevant is shown in the figure below

65. C1C1 C2C2 C3C3 C1C1 C2C2 C3C3 Pearlite a micro-constituent (Not a phase) Similar* to C 1 but the proeutectoid phase is Cementite *Similar in solidification sequence – not in properties Pro-eutectoid Cementite along prior Austenite grain boundaries Fe 3 C  Grain boundary

66. SEM micrograph of pearlite in Eutectoid composition (0.8%C) steel  Pearlite is a micro-constituent with alternating lamellae of cementite and ferrite. SEM micrograph of pearlite in Hyper-eutectoid composition (1.0%C) steel Pro-eutectoid Cementite along prior Austenite grain boundaries Pearlite

67.  Composition of EN9 alloy in wt.%: C-0.5, Si-0.25, Mn-0.70, S-0.05, P-0.05  Processing: Hot rolled steel sample annealed at 700  C and followed by slow cooling  Hardness: 180-230 BHN (Avg. = 205 BHN)  Images taken on Solver Pro NT-MDT- semi contact mode. Structure of Etched Pearlite in AFM Images courtesy: Prof. Sandeep Sangal and K. Chandra Sekhar (MSE, IITK).

68. Funda Check  Components need not be only elements- they can be compounds like Al 2 O 3, Cr 2 O 3.  Phase diagrams usually do not correspond to the global energy minimum- hence often microstructures are ‘tolerated’ in phase diagrams.  Phase diagrams give information on stable phases expected for a given set of thermodynamic parameters (like T, P). E.g. for a given composition, T and P the phase diagram will indicate the stable phase(s) (and their fractions).  Phase diagrams do not contain microstructural information- they are often ‘overlaid’ on phase diagrams for convenience.  Metastable phases like cementite are often included in phase diagrams. This is to extend the practical utility of phase diagrams.  Strictly speaking ‘cooling curves’ (curves where T changes) should not be overlaid on phase diagrams. (Again this is done to extend the practical utility of phase diagrams  assuming that the cooling is ‘slow’).

69. Solved Example Two separate alloys are cooled from the  phase field: (i) one hyper-eutectoid and other (ii) hypoeutectoid. Assuming that in both cases the pro-eutectoid phase forms along the  grain boundaries and its phase fraction is 6%, determine the carbon composition of the two alloys. %C → T → Fe 0.8 C1C1   + Fe 3 C 723ºC 0.025 RT~0.008 C2C2 At 723  C

70. End

71. Solved Example Data for Cu:   H f (Cu vacancy) = 120  10 3 J/mole  R (Gas constant) = 8.314 J/mole/K

72. A B %B → T → M.P. of B M.P. of A C = 1 P = 2 F = 0 C = 2 P = 1 F = 2 C = 2 P = 1 F = 2 Variables → T, C S B  2 Variables → T, C L B  2 Variables → T, C L B, C S B  3 2 – P 3 – P C = 2 P = 2 F = 1 2 – P F = C – P + 1 Say T 1 is chosen The compositions C L B & C S B are automatically chosen by the system

73. Cooling PbSn 100 200 300 1030 507090 L   +   L +   + L Hypereutectic composition  Proeutectic   +  Eutectic Mixture of   Grain boundary Liquid (melt)

74. Cooling C1C1 C2C2 %C → T → 0.8 0.02 Eutectoid  →  + Fe 3 C    + Fe 3 C ++  + Fe 3 C Fe C3C3

75. T (ºC) → 900 1300 1500 AgPt %Pt → 1030 507090 L   +   L +  1100 1700 Melting points of the components vastly different Peritectic reaction L +  → 