1. ALLOYS & PHASE DIAGRAMS

2. Jean P. Mercier, Gérald Zambelli, Wilfried Kurz, Introduction à la Science des Matériaux, Presses Polytechniques et Universitaires Romandes, ISBN
Jean P. Mercier, Gerald Zambelli, Wilfried Kurz, Introduction to Materials Science, Elsevier, ISBN
Pure materials often have properties of technologically marginal interest.
Most materials are mixtures of atoms or molecules of different nature. In some cases, the constituents (atoms or molecules) of the material are soluble in any proportion and the mixtures are homogeneous. In other cases, the components are only partially miscible. The material then contains multiple phases of different composition and structure.
The combination of these different phases produced microstructures that dramatically influence the properties and specifications of materials.
Phase diagrams are the basis for analyzing the formation of microstructures. They define in a simple manner the equilibrium states between phases.

3. (over the whole compositional range)
Full miscibility A B
Solid B
(over the whole compositional range)
Solid solution
Solid A

4. Limited miscibility A Two phases (Miscibility Gap) Solid solution
(A in B) A B
Solid B
Two phases
(Miscibility Gap)
(Solvus)
Solid A
Solid solution
B in A

5. Limited miscibility Intermediate compound Intermediate compound
Solid solution
(A in B) A B
Solid B
Solid A
Intermediate compound
Solid solution
B in A

6. Alloy
Conventionally, an alloy is a material with metallic character combined with one or more metallic elements and/or possibly elements with non-metallic character. This concept, which is currently evolving, is gradually extended to other materials (ceramics, polymers). We now defines an alloy as a combination of atoms, ions or molecules to form a material whose properties differ from those of components. An alloy is usually composed of several phases of microscopic size having different compositions and structures that contribute synergistically to the material properties.

7. An alloy is generally composed of two or more components and includes one or more phases.
A phase is a part of the system wherein the composition (type and concentration of the components present) and the atomic organization (crystalline or amorphous structure ...) are fixed.
In a system at (thermodynamic) equilibrium, each phase has a fixed composition which is homogeneous in its full extent. (In non-equilibrium systems, very frequently encountered in materials, the composition may vary depending on time and location inside the material.)
Each phase is separated from other phases of the system by an interface.
For instance: the amorphous and crystalline phases are distinguished by their different atomic arrangements. Similarly, the component iron may exist in the vapor state, liquid state, cubic face-centered crystalline solid state or cubic centered crystal representing four different phases.

9. Examples
A well known example of a metal alloy having two phases is that of steels or carbon steels.
Pure Cu has a very high electrical conductivity, but its mechanical strength is not sufficient when the mechanical stresses are high. In order to harden it, another component is added, for example Sn or Zn.
By adding a certain amount of Sn to Cu, a bronze is obtained which is an alloy consisting of two phases: Cu + CuxSny.
By adding a certain amount of Zn to Cu, brass is obtained which is an alloy consisting of two phases: Cu + CuxZny.
A "hard metal" alloy used for cutting tool normally contains three components, Co, W and C, forming two phases: tungsten carbide (WC), in the form of grains, and a metallic Co, which is the ductile matrix linking hard and fragile tungsten carbide grains.
Ruby used for the construction of lasers is a single crystalline alloy of
Al2O3 (sapphire) and 0.05 percent Cr2O3. The characteristic red color of this crystal is due to the presence of Cr3+ ions dispersed in the sapphire crystal as a solid solution. In this case, the alloy has only one phase although it has two components.

10. Interfaces between phases A phase occupies a determined volume in space limited by an interface that separates it from another phase. The grain boundaries are a particular type of interface as they mark the boundary between two crystal grains of the same composition and the same structure, but with different orientation. In general, the creation of an interface is a process that requires some work and which is therefore not energetically advantageous from the point of view of thermodynamics. The presence of interfaces therefore increases the free energy of a material. In general, fine grained systems, which are characterized by the presence of many interfaces, have a higher free energy than coarse systems. From the thermodynamic point of view, an interface is characterized by the specific interfacial energy 𝛾 [Jm-2], which, under certain conditions, can be equated to a surface tension [Nm-1]. The interfacial energy is independent of the interface area. It can be determined by measuring the work required to create a unitary interface.

11. Phase diagram of a pure crystallizable substance
ONE COMPONENT SYSTEMS
Phase diagram of a pure crystallizable substance
Generally, the temperature of the melting point increases very moderately with pressure.
There are few substances such as water, Bi and Ga, which exhibit an increase in volume during crystallization.
In this case, there is a lowering of the melting point temperature when the pressure is raised.
Phase diagram at equilibrium for a pure substance describing the areas of stability of the crystalline solid, liquid and vapor. T = triple point, C = critical point, Tm (P1) = melting temperature at pressure P1 and Tv (P1) = vaporization temperature at pressure P1.

12. Gibbs Phase Rule
Gibbs phase rule determines the number of phases p present in a system at equilibrium:
p + f = n + 2
where n represents the number of components in the system and f the number of degrees of freedom of the system, i.e. the number of independent intensive variables.
If we consider a system at equilibrium, there is a total of n + 1 variables to be determined (n - 1 mole fractions, T and P). A number of these variables can be set arbitrarily without changing the physical state of the system. This is called the number of degrees of freedom or the variance f of the system.
In systems where pressure is fixed, the variance is automatically reduced by one:
f + p = n + l (P = const) .

13. Phase diagram of a one component pure crystallizable substance
Phase diagram at equilibrium for a pure substance describing the areas of stability of the crystalline solid, liquid and vapor. T = triple point, C = critical point, Tm (P1) = melting temperature at pressure P1 and Tv (P1) = vaporization temperature at pressure P1.

14. Let us apply the phase rule to the phase diagram of a pure substance
For a pure substance, n = 1. The sum of the variance f and of the number of phases p is calculated and is always equal to three.
In other words, the variance of a system consisting of a pure substance and having three phases in equilibrium, is equal to zero. Therefore in the equilibrium phase diagram, there is only one point (the triple point T' characterized by a single value of P and T) for which there is simultaneously the presence of three phases at equilibrium.
The (one dimensional) curves O'T'B and T'C determine the temperatures and pressures at which two phases can coexist. In this case, the variance f is unitary, that is to say that for the points of the system located on one of these three curves, one can choose freely either the temperature or the pressure (the second variable being determined by the first choice).
The areas between the curves (OT'-T'B, T'B-T'C for example) are regions of the phase diagram where there is - at equilibrium - only the presence of one phase.

15. The most stable phase always has the smallest free enthalpy.
The free energy G of a phase is connected to its enthalpy H and its entropy S by the well known thermodynamic relationship (Smith, 1990): G = H - TS
As the entropy S is higher for less ordered phases, their free enthalpy G decreases more rapidly with temperature than that of the more ordered phases.
The most stable phase always has the smallest free enthalpy.
Variation - at constant pressure - of the free enthalpy of the three stable phases of a pure substance as a function of temperature. Point A corresponds to the melting point Tm, B to the boiling point or vaporization point Tv, and C to a hypothetical transition between solid and vapor. The upper part of the curves not in bold correspond to thermodynamic states out-of-equilibrium.

16. Intermediate compound
BINARY SYSTEMS
PHASES EQUILIBRIUM IN BINARY SYSTEMS
Let us examine the conditions for which a mixture of two components in equilibrium form a solution (solid or liquid), or exists as two distinct (liquid or solid) phases. A B
Solid A
Solid B
Two phases
Solid solution
B in A
(A in B)
(Miscibility Gap)
(Solvus)
Intermediate compound

17. Binary systems ? Mixture of 2 phases A & B1 A B
Molar fraction XB
Molar free entalphy G
T, P = cste
Gm
Mixture of 2 phases A & B over the whole range of composition ? 1 A B
Molar fraction XB
Molar free entalphy G
T, P = cste
Gm
∆Gm
Solid solution AB over the whole range of composition 1 A B
Molar fraction XB
Molar free entalphy G
T, P = cste
Gm
∆Gm
Variation of the free enthalpy of mixing Gm for a phase a consisting in two elements A and B, at constant T and P, in function of the molar fraction of element B.

18. Binary systems Two phases: solid & liquid One phase
Variation of the free enthalpy of mixing Gm for a phase a consisting in two elements A and B, at constant T and P, in function of the molar fraction of element B.
Variation of the free enthalpy G of a binary system at constant T and P:
(a) enthalpy of mixing of a binary system A,B with a solid phase a and a liquid phase l.
Gm and Gml are the free enthalpy of the phase  and l, respectively. The 2 curves cross at G0, corresponding to the concentration X0,
(b) Part of the diagram corresponding to a heterogeneous mixture (of two alloys of composition Xe and Xel.

19. Binary systems What happens when the temperature changes?
The relative quantity of the two phases is determined by the lever rule:
f and fl are the proportion of the  and l phase, respectively.
Since f + fl = 1, it is easy to demonstrate that:
Binary system with two fully miscible components A and B.
(a) to (f) Variation of the free enthalpy of mixing at various temperatures;
(g) Phase diagram at equilibrium

20. Binary systems
The relative quantity of the two phases is determined by the lever rule:
f and fl are the proportion of the  and l phase, respectively.
Since f + fl = 1, it is easy to demonstrate that:
Binary system with two fully miscible components A and B.
(a) to (f) Variation of the free enthalpy of mixing at various temperatures;
(g) Phase diagram at equilibrium

21. Binary systems
The relative quantity of the two phases is determined by the lever rule:
f and fl are the proportion of the  and l phase, respectively.
Since f + fl = 1, it is easy to demonstrate that:
Binary system with two fully miscible components A and B.
(a) to (f) Variation of the free enthalpy of mixing at various temperatures;
(g) Phase diagram at equilibrium

22. Binary systems
The relative quantity of the two phases is determined by the lever rule:
f and fl are the proportion of the  and l phase, respectively.
Since f + fl = 1, it is easy to demonstrate that:
Binary system with two fully miscible components A and B.
(a) to (f) Variation of the free enthalpy of mixing at various temperatures;
(g) Phase diagram at equilibrium

23. Binary systems
The relative quantity of the two phases is determined by the lever rule:
f and fl are the proportion of the  and l phase, respectively.
Since f + fl = 1, it is easy to demonstrate that:
Binary system with two fully miscible components A and B.
(a) to (f) Variation of the free enthalpy of mixing at various temperatures;
(g) Phase diagram at equilibrium

24. Binary systems
The relative quantity of the two phases is determined by the lever rule:
f and fl are the proportion of the  and l phase, respectively.
Since f + fl = 1, it is easy to demonstrate that:
Binary system with two fully miscible components A and B.
(a) to (f) Variation of the free enthalpy of mixing at various temperatures;
(g) Phase diagram at equilibrium

25. Binary systems
The relative quantity of the two phases is determined by the lever rule:
f and fl are the proportion of the  and l phase, respectively.
Since f + fl = 1, it is easy to demonstrate that:
Binary system with two fully miscible components A and B.
(a) to (f) Variation of the free enthalpy of mixing at various temperatures;
(g) Phase diagram at equilibrium

26. Binary systems
The relative quantity of the two phases is determined by the lever rule:
f and fl are the proportion of the  and l phase, respectively.
Since f + fl = 1, it is easy to demonstrate that:
Binary system with two fully miscible components A and B.
(a) to (f) Variation of the free enthalpy of mixing at various temperatures;
(g) Phase diagram at equilibrium

27. Alloys - Miscibility Gap Experimental construction of a phase diagram
Binary systems
Experimental construction of a phase diagram
When a system consist in 2 components, it can form either a solid solution in the region of miscibility, or split into 2 phases (phase mixture)in the miscibility gap.
Variation of the free enthalpy of a systems of 2 components A and B exhibiting a miscibility gap at various temperatures
Equilibrium phase diagram

28. Alloys - Miscibility Gap Experimental construction of a phase diagram
Binary systems
Experimental construction of a phase diagram
When a system consist in 2 components, it can form either a solid solution in the region of miscibility, or split into 2 phases (phase mixture)in the miscibility gap.
Variation of the free enthalpy of a systems of 2 components A and B exhibiting a miscibility gap at various temperatures
Equilibrium phase diagram

29. Solid Solutions & Phase Mixture
A solid solution is a single phase which exists over a range in chemical compositions.
Almost all minerals are able to tolerate variations in their chemistry (some more than others).
Chemical variation greatly affects the stability and the behaviour of the mineral.
Therefore it is crucial to understand:
the factors controlling the extent of solid solution tolerated by a mineral
the variation in enthalpy and entropy as a function of chemical composition
different types of phase transition that can occur in solid solutions

30. Example: Olivines Mg2SiO4 Fe2SiO4
The two most important forms of olivine are:
Forsterite
Mg2SiO4
Fayalite
Fe2SiO4
and
Both forms have identical structures and identical symmetry. They differ only in the type of cation occupying the M sites.
Most natural olivines contain a mixture of Mg and Fe on the M sites: (Mg, Fe)2SiO4. This is an example of substitutional solid solution.
The composition of the solid solution is specified in terms of the mole fraction of the two endmembers forsterite and fayalite.
e.g. (Mg0.4Fe2+0.6)2SiO4 is said to contain 40% forsterite and 60% fayalite (often abbreviated to Fo40Fa60)

31. Substitutional solid solutions
Specific sites in the structure (e.g. M-sites in olivine) are occupied by either Mg or Fe.
In the ideal case, Mg and Fe are randomly distributed.
The probability of any one site being Mg is equal to the mole fraction of Mg in the system.
e.g. in Fo40Fa60, each site has a 40% chance of being occupied by Mg and a 60% change of being occupied by Fe.

32. Other types of solid solution
Coupled substitution : Cations of different charge are substituted for each other. Requires two coupled substitutions to maintain charge balance. e.g. Al3+ + Ca2+ = Si4+ + Na+ in plagioclase feldspars.
[NaAlSi3O8 Albite (Ab) & CaAl2Si2O8 Anorthite (An)]
Omission solid solution : Chemical variation achieved by omitting cations from sites that are normally occupied. e.g. pyrhottite solid solution between FeS and Fe7S8. Charge balance is achieved by changing the valance of transition metal cations (e.g. Fe2+ is converted to Fe3+)
Interstitial solid solution : Cations are inserted into sites not normally occupied in the structure. e.g. solid solution between tridymite (SiO2) and nephaline (NaAlSiO4) achieved by stuffing Na into channel sites and substituting Al for Si in framework.

33. Factors controlling the extent of solid solution
Cation Size
If cation sizes are very similar (i.e. ionic radii differ by less than 15%) then extensive or complete solid solution is often observed. Mg
0.86 Å
Fe2+
0.92 Å Ca
1.14 Å 7%
32%

34. Factors controlling the extent of solid solution
Crystal Structure & Structural flexibility
Complete miscibility implies the same crystal structure
Cation size alone is not enough to determine the extent of solid solution, it also depends on the ability of the structural framework to flex and accommodate differently-sized cations
e.g. there is extensive solid solution between MgCO3 and CaCO3 at high temperature
Règles de Hume- Rothery (empiriques)
Same size of the cations (atoms)
Same (crystal) structure
Same charge or valence (max. differencce tolerated -+ 1 when charge balance can be preserved)
Cation charge (Valence)
Complete solid solution is usually only possible if the substituting cations differ by a maximum of ± 1. Heterovalent (coupled) substitutions often lead to complex behaviour at low temperatures due to the need to maintain local charge balance.
[Règles de Hume-Rothery (empiriques)]

35. Factors controlling the extent of solid solution
Crystal Structure & Structural flexibility
Règles de Hume- Rothery (empiriques)
Same size of the cations (atoms)
Same (crystal) structure
Same charge or valence
(max. difference tolerated -+ 1 when charge balance can be preserved)
Complete miscibility implies the same crystal structure
Cation size alone is not enough to determine the extent of solid solution, it also depends on the ability of the structural framework to flex and accommodate differently-sized cations
e.g. there is extensive solid solution between MgCO3 and CaCO3 at high temperature
Règles de Hume- Rothery (empiriques)
Same size of the cations (atoms)
Same (crystal) structure
Same charge or valence (max. differencce tolerated -+ 1 when charge balance can be preserved)
Cation charge (Valence)
Complete solid solution is usually only possible if the substituting cations differ by a maximum of ± 1. Heterovalent (coupled) substitutions often lead to complex behaviour at low temperatures due to the need to maintain local charge balance.

36. Thermodynamics of solid solutions
(Phase mixture)
Exsolution
Cation ordering

37. Factors controlling the extent of solid solution
Temperature
Extent of solid solution tolerated is greater at higher temperatures
G = H – TS
Cation-size mismatch increases the enthalpy (structure must strain to accommodate cations of different size): solid solution is destabilised at low temperatures
Cation disorder in a solid solution increases the configurational entropy: solid solution is stabilised at high temperature

38. Thermodynamics of solid solutions
Enthalpy
H = ½ N z (xA2 WAA + xB2 WBB
+ 2 xA xB WAB)
N = number of sites, z = coordination
xA,B = molar fraction of A,B
H = H(mechanical mixture) + DH(mixing) =
½ N z (xA WAA + xB WBB) +
½ N z xA xB (2WAB - WAA - WBB )
½ N z (xA WAA + xB WBB) + ½ N z xA xB (2WAB - WAA - WBB ) = ½ N z (xA WAA + xB WBB)[(xA+xB)=1] + ½ N z xA xB (2WAB - WAA - WBB )
J = (2WAB - WAA - WBB )
If J > 0 then like neighbours (AA and BB) are favoured over unlike neighbours (AB)
If J < 0 then unlike neighbours (AB) are favoured over like neighbours (AA and BB)

39. Thermodynamics of solid solutions
Enthalpy
H = ½ N z (xA2 WAA + xB2 WBB
+ 2 xA xB WAB)
½ N z (xA WAA + xB WBB) + ½ N z xA xB (2WAB - WAA - WBB ) = ½ N z (xA WAA + xB WBB)[(xA+xB)=1] + ½ N z xA xB (2WAB - WAA - WBB )
N = number of sites, z = coordination
xA,B = molar fraction of A,B
H = H(mechanical mixture) + DH(mixing) =
½ N z (xA WAA + xB WBB) +
½ N z xA xB (2WAB - WAA - WBB )
½ N z (xA WAA + xB WBB) + ½ N z xA xB (2WAB - WAA - WBB ) = ½ N z (xA WAA + xB WBB)[(xA+xB)=1] + ½ N z xA xB (2WAB - WAA - WBB )
J = (2WAB - WAA - WBB )
If J > 0 then like neighbours (AA and BB) are favoured over unlike neighbours (AB)
If J < 0 then unlike neighbours (AB) are favoured over like neighbours (AA and BB)

40. Thermodynamics of solid solutions Enthalpy
Hmix = 1/2 Nz xAxB [2WAB - WAA - WBB]
= 1/2 Nz xAxB J
If J > 0 then like neighbours (AA and BB) are favoured over unlike neighbours (AB)
If J < 0 then unlike neighbours (AB) are favoured over like neighbours (AA and BB)

41. Thermodynamics of solid solutions
Entropy
DSmix = k ln w
w = no. of degenerate configurations of the system
For a collection of N atoms, consisting of NA A atoms and NB B atoms, the no. of configurations is:
Stirling formula: lnN! = NlnN-N (when N very large)
Simplifies to:
DSmix = - R (xA ln xA + xB ln xB)
per mole of sites.
Stirling formula: lnN! = NlnN-N (when N very large)

42. Gmix = Hmix - TSmix Free energy of mixing Hmix = 1/2 Nz xAxB W
Smix = - R (xA ln xA + xB ln xB)
Gmix = Hmix - TSmix

43. Ideal solid solution (DHmix = 0)
In an ideal solid solution, the two cations substituting for each other are very similar (same charge and similar size).
Gmix = - TSmix
Solid solution is stable at all compositions and all temperatures.
Example:
The forsterite-fayalite solid solution is very close to ideal!

44. Non-ideal solid solution (DHmix < 0)
Obviously we have a solid solution over the whole composition range.
However, here the change in free energy relative to the ideal case is more subtle There is a strong driving force for cation ordering in the centre of the solid solution, where the ratio of A:B cations is 1:1. The fully-ordered phase has zero configurational entropy, because there is just one way to arrange the atoms (two if you include the equivalent anti-ordered state). However, it has a low enthalpy due to the energetically-favourable arrangement of cations.
This stabilises the ordered phase at low temperature, where the –TΔSmix term in the free energy is less important. The fully disordered solid solution has a high configurational entropy, which stabilises it at high temperature.

45. Non-ideal solid solution (DHmix < 0)
The ordered phase is stabilized at low temperature, where the –TΔSmix term in the free energy is less important.
The fully disordered solid solution has a high configurational entropy, which stabilises it at high temperature.
There is a phase transition from the ordered phase to the disordered phase at a critical transition temperature (Tc). The phase diagram for a 2nd-order cation ordering phase transition simply shows a transition from a single-phase disordered solid solution to a single-phase ordered solid solution at each composition, with the transition temperature varying parabolically as a function of composition.
Cation ordering phase transition occurs below a particular
transition temperature Tc
Tc varies with composition - highest in the centre when A:B ratio is 1:1

46. Non-ideal solid solution (DHmix > 0)
Free energy develops minima at low temperature due to competition between positive DHmix term and negative -TDSmix term.
Solid solution is unstable at intermediate compositions.
Equilibrium behaviour is given by the common tangent construction.

47. Non-ideal solid solution (DHmix > 0)
Phase mixture:
Phase with bulk composition C0 splits into two phases Q and R with compositions C1 and C2.
Lever Rule:
Proportion of Q
= (C2-C0)/(C2-C1)
= b / (a+b)
Proportion of R =
(C0-C1)/(C2-C1)
= a / (a+b) a b C0

48. Crystallisation of an ideal solid solution from a melt

49. Crystallisation of an ideal solid solution from a melt

50. Equilibrium vs. fractional crystallisation
Equilibrium crystallisation
When cooling is very slow the liquid and melt remain in equilibrium with each other. The compositions of the phases are able to continuously adjust to the new temperature. Final result is a homogeneous solid with the same composition as original melt (if cooling too quick get zoning).
Fractional crystallisation
Crystals are removed from contact with the melt as soon as they form. Since crystals forming are enriched in one element, the residual melt becomes progressively enriched in the other element. The bulk composition of the system evolves with time.

51. Relationship between the thermal analysis curves and the phase equilibrium diagram in case of a completely miscible binary system
(a) thermal analysis curves
(b) Temperature-time diagram for XB = 0, 0.5 and 1s
(c) The cooling curve does not show a plateau for the phase change due to the existence of a range of transformation l-a
(After Eisenstadt, 1971)

52. Other types of melting loops
If the curvature of the G-x curves for solid solution and melt are very different, then melting loops of the form shown here may result.
In the first case, the curvature of the melt G-x curve is greater than that of the solid solution, and both liquidus and solidus show a thermal minimum, with melting loops on either side. At the minimum, melt with intermediate composition crystallises directly into a solid solution, without going through a two-phase region.
If the curvature of the solid solution G-x curve is greater than that of the melt, then a thermal maximum arises and the loops are inverted.

53. Partial miscibility in the solid state: system with a melting point minimum
When the free enthalpy diagram of variation of the crystalline phase in function of the composition has two minima separated by a maximum, the corresponding phase diagram exhibit a miscibility gap in the solid state. In this case, it is genrally observed that the solid-liquid equilibrium curve has a minimum melting point. The liquidus and solidus curves then have an intermediate contact point having a horizontal tangent (Au-Ni system).
At a concentration Xm, the solid solution, and the liquid have the same composition, the liquidus temperature is the same as the solidus temperature and the thermal analysis curve that is characterized by a plateau, is similar to that of a pure substance.
Note that at the minimum, the temperatures of the liquidus and solidus are identical.

54. Crystallisation of two endmembers with limited solid solution
We now explore the phase diagrams of systems where the two endmembers have different structures and show only limited solid solution. Because the two endmembers have different structures, they are represented by separate free energy curves (labelled and ). Because the degree of solid solution permitted in each endmember is limited, the curvature of the free energy curves is much larger than depicted so far.
T = T1. Melt is stable at all compositions.
T = TA. Crystallisation temperature of pure A.
T = TB. Crystallisation temperature of pure B. Two-phase region in A-rich compositions.
T = TE. At one particular temperature, all three phases lie on the same common tangent (i.e. are in equilibrium with each other). This is called the eutectic temperature.
T = T2. Below the eutectic temperature the melt is no longer stable at any composition, and there is a large central miscibility gap.
The resulting phase diagram can be interpreted as the intersection of a melting loop of the form shown prviously with a simple solvus of the form shown in Fig. 9. The eutectic point defines the composition of the melt in equilibrium with two solids at the eutectic temperature.

55. Construction of a eutectic diagram for two components A et B that crystalize in two different crystal structures a et b
shows the spindle-shaped diagram of the components A et B when they crystalize in the structure a
shows the spindle-shaped diagram of the components A et B when they crystalize in the structure b
The solubility domains of the solid solutions B within A assuming the structure a and A within B assuming the structure b.
note that part of the diagrams is metastable (as indicated with dash lines)
Combining the equilibrium diagrams a, b and c (only stable parts) gives the eutectic equilibrium phase diagram.
(e) Three phases equilibrium at the eutectic temperature : lE = aE + bE.

56. The silver-copper system is an example of a eutectic system (Fig. 8
It is noted that the ‘T-t’ cooling curves 1 and 4 of the pure Ag phase and that of the eutectic alloy are of the same type, namely they exhibit a plateau where the temperature does not change during the whole transformation. In contrast the cooling curve 2 of a non-eutectic composition exhibit only a slowing down of the cooling (Fig 8.19b). For compositions between the maximum solubility points of  and  (curve 3 and 5), the curve exhibit first a slowing down of the cooling, followed by a plateau of variable duration where the temperature doesn’t change and where the residual liquid transforms into a biphasic solid of eutectic composition.
Fig 8.19 Example of a eutectic alloy: (a) equilibrium phase diagram of the Ag-Cu system, et (b) ‘T-t’ cooling curves for compositions 1 to 5 labeled on the phase diagram: (1) Polycrystal of pure  (Ag). (2) Polycrystal of  solid solution (3) solid solution of  crystals enclosed in biphasic + eutectic, (4) solid solution of  crystals enclosed in biphasic + eutectic

57. Crystallisation of two endmembers with no solid solution
In the limit that no solid solution is tolerated by end members A and B, the phase diagram appears

58. Binary peritectic Diagram
Binary phase diagram with a peritectic point built from the various phase diagrams of the considered phases
The peritectic diagram (d) is built combining the three diagrams (a), (b) et (c).
(e) Three phase equilibrium at the peritectic temperature:
Tp: lp + ap = bp.

59. Binary systems with intermediate compounds
Intermediate compound AB2 splits the phase diagram into two binaries.
Here AB2 melts congruously, i.e. it melts to give a liquid with the same bulk composition

60. Incongruent melting and peritectic points
An incongruent melting compound melts to give a mixture of solid and melt of different compositions X

61. Example: Forsterite-Quartz
Mg2SiO4
Fosterite
MgSiO3
Enstatite
SiO2
Quartz
Cristobalite
Tridymite

62. Equilibrium phase diagram for the copper -magnesium system with formation of two intermetallic compounds, MgCu2 et Mg2Cu
Equilibrium phase diagram for the copper -magnesium system with formation of two intermetallic compounds, MgCu2 et Mg2Cu. The concemtration XMg is given in molar fraction.

64. Iron - Cementite (Fe3C) diagram
Iron - Cementite (Fe3C) phase diagram: (a) metastable diagram used for steels and white iron; (b) The delta region around the peritectic point of the phase diagram, (c) The region around the eutectoid point of the phase diagram. The iron concentration is given in weight %.

65. Typical Iron-Iron Carbide Applications
The purpose of adding carbon to iron is to change the properties of the iron. Carbon is a very hard substance and when it is added to iron, it becomes a much stronger material, depending on the amount of carbon added. Pure iron itself is relatively soft and ductile compared to carbon impregnated steels. By adding carbon to the iron you get a variety of strengths in your steel.
From the example of iron carbide, you may have noticed the percent carbon that is included in a specimen of iron. Iron has allotropic properties. This means it acquires a different crystal structure at different temperatures. Iron will transform to a BCC [more strength] structure or ferrite at cooling temperatures and then go to FCC [more ductile] at a lower temp also called austenite. Cementite is the name for iron carbide. It defines the point where the solubility of carbon in iron is at its maximum. This will yield a very hard and brittle steel which may not be suited for all applications.
Let's take a look at low (or mild), medium, and high carbon steels and see what conditions they are best suited for.
Low Carbon Steels Applications for low carbon steel are seen in joists in building construction, body panels for automobiles, wire and nails. These fabrications adapt low carbon steel for good reason. The low cost of production of this steel is beneficial because of the number of uses it supplies. Wire and nails are found almost anywhere something is built and there will always be a need for body panels on cars to be made. The ductility of low carbon steel allows for ease of manipulation and reparability (i.e. dents in car fenders, bending wire).
Medium Carbon Steels These steels will have a higher tensile strength and less ductility than mild steel. Applications for this include farm equipment, engine components, gears, and structural fixtures. Farm equipment and engine components are parts that you want to be very strong and durable yet not brittle. The medium carbon content allows the strength to be there and will hold up to many cycles of stress and strain. In an engine, the connecting rods and crankshaft endure a lot of loading and unloading. Too brittle a material could result in mechanical failures over time. And because they will be subject to high temperatures, too soft a steel will result in extreme elongation. The higher carbon content reduces elongation at high temperatures.

66. Figure: This image shows a layer of carbon applied toa gear
Figure: This image shows a layer of carbon applied toa gear. This is called "case hardening" which can be done by a process of heat treatment and diffusion. The reason for this is to have great wear resistance on the outside of the gear where it interfaces with another gear. The center portionis basic steel which retains good strength properties and will not be subject to cracking.
High Carbon Steels This form of steels has the best hardness, strength, and least ductility. The areas best suited for this steel is in tools, drills, saws, knife blades, and bearings. High carbon content reduces the wear and deformation of the steel. Many types of tools require this because you don't want to replace a hammer after three hits on a nail or wear out a drill bit too soon. Knife blades and axes follow the same principal. For these applications, the harder the steel, the better.

67. Iron-Iron Carbide Phase Diagram Example
Allotropic changes take place when there is a change in crystal lattice structure. From 2802º-2552ºF the delta iron has a body-centered cubic lattice structure.
At 2552ºF, the lattice changes from a body-centered cubic to a face-centered cubic lattice type.
At 1400ºF, the curve shows a plateau but this does not signify an allotropic change. It is called the Curie temperature, where the metal changes its magnetic properties.
Two very important phase changes take place at 0.83%C and at 4.3% C. At 0.83%C, the transformation is eutectoid, called pearlite.
gamma (austenite) --> alpha + Fe3C (cementite)
At 4.3% C and 2066ºF, the transformation is eutectic, called ledeburite.
L(liquid) --> gamma (austenite) + Fe3C (cementite)
The Figure shows the equilibrium diagram for combinations of carbon in a solid solution of iron. The diagram shows iron and carbons combined to form Fe-Fe3C at the 6.67%C end of the diagram. The left side of the diagram is pure iron combined with carbon, resulting in steel alloys. Three significant regions can be made relative to the steel portion of the diagram. They are the eutectoid E, the hypoeutectoid A, and the hypereutectoid B. The right side of the pure iron line is carbon in combination with various forms of iron called alpha iron (ferrite), gamma iron (austenite), and delta iron. The black dots mark clickable sections of the diagram.

68. Stainless steel phase diagram at 900 degrees Celsius (ASM 1-27)
Ternary phase diagram
A Ternary Diagram Plot is used in chemistry for depicting chemical compositions - phase diagrams. These diagrams are three-dimensional but is illustrated in two-dimensions for ease of drawing and interpretation. In a ternary diagram the relative percentage (normally weight %) of three components are represented by A, B and C. The only requirement is that the three components have to sum to 100%. If they don't, you have to normalize them to 100%.
18% Cr
8% Ni
(74 % Fe)
Stainless steel phase diagram at 900 degrees Celsius (ASM 1-27)

69. How do you read a ternary phase diagram?C A B 20 40 60 80 C A B 50 40 10
50% A
10% B
40 % C

70. Ternary phase diagram (2)
Stainless steel solidus projections over a range of temperatures (ASM 3-44)
This Figure represents the entire ternary phase diagram solidus projections of stainless steel over a range of temperatures under constant pressure. This picture illustrates how temperature affects solubility.

71. Ternary phase diagram (3)

72. Ternary phase diagram (4)

73. Documents (in particular: Assignment) 8/
8/PhaseDiagrams-2017-English.ppt
8/Assignment-chap8.doc
8/Exercices-chap8.doc