PHASE DIAGRAMS
Phase – a chemically and structurally homogenous region of a material. Region of uniform physical and chemical characteristics. Phase boundaries separate two distinct phases.
A single phase system is called homogeneous. A system with two or more phases is called heterogeneous.
Phase Diagram – a graphic representation showing the phase or phases present for a given composition, temperature and pressure.
Component – the chemical elements which make up the alloy.
Solvent atoms: primary atomic species. Host atoms Solute atoms: the impurities. Normally the minor component Solubility Limit ‐ Maximum concentration of solute atoms that may dissolve in the solvent to form a solid solution. The excess
of solute forms another phase of different composition. Example: water‐sugar
Phase Diagrams of Pure Substances
• Predicts the stable phase as a function of Ptotal and T. Example: water can exist in solid, liquid and vapor phases, depending on the conditions of temperature and pressure.
• Characteristic shape punctuated by unique points.
– Phase equilibrium lines
– Triple Point (three different phases of water in equilibrium)
– Critical Point
Example:
In the pressure‐temperature (PT) phase diagram of water there exists a triple point at low pressure (4.579 torr) and low temperature (0.0098oC) where solid, liquid and vapor phases of water coexists.
Vaporization Line – Liquid and vapor coexists
Freezing Line – Liquid and solid coexist.
Sublimation Line – Solid and vapor coexist
Phase ‐ Any portion including the whole of a system, which is physically homogeneous within it and bounded by a surface so that it is mechanically separable from any other portions.
Gibbs Phase Rule
From thermodynamic considerations, J.W. Gibbs (1839‐1903 American physicist – University of Yale) derived the following equation:
P + F = C + 2
Where
P = number of phases which coexists in a given system
F = degrees of freedom
C = number of components in the system
2 = one can vary temperature and pressure
F = 0 zero degrees of freedom. Neither P or T can be change (a point –
invariant point)
F = 1 one degree of freedom. One variable (P or T) can be changed independently (a line)
F = 2 two degrees of freedom. Two variables (P or T) can be changed independently (an area).
Schematic unary phase diagram for magnesium, showing the melting and boiling temperatures at one atmosphere pressure.
C= 1 for pure magnesium
Point A:
P= 1 for pure liquid phase
2+C=F+P
2+1=F+1
F=2 degrees of freedom – change
pressure and temperature in liquid
Phase. Point B: P= 2 for liquid and solid
2+C=F+P
2+1=F+2
F=1 degrees of freedom – change pressure or temperature (and the other variable is dependent – to stay on the line).
Point X:
P= 3 (liquid, solid and vapor coexist)
2+C=F+P
2+1=F+3
F=0 degrees of freedom – pressure and temperature are fixed at the the single point called the triple point.
Example
‐ For pure substance where P and T can be changed P + F = C + 2 = 1 + 2 = 3
Pure substance in a triple point, then C = 1 (one component)
and P = 3 (number of phases that coexist)
The value of F is zero (zero degrees of freedom) the three phases coexist in a point.
‐ For pure substance where P and T can be changed P + F = 1 + 2 = 3
Pure substance in a freezing line, then C = 1 (one component)
and P = 2 (number of phases that coexist)
The value of F is one (one degree of freedom) the two phases (solid and liquid) coexist in a line.
Solubility: The amount of one material that will completely dissolve in a second material without creating a second phase.
Unlimited solubility: When the amount of one material that will dissolve in a second material without creating a second phase is
unlimited.
Limited solubility ‐ When only a maximum amount of a solute material can be dissolved in a solvent material.
Solid Solution:
Solid‐solution strengthening ‐ Increasing the strength of a metallic material via the formation of a solid solution.
Dispersion strengthening ‐ Strengthening, typically used in metallic materials, by the formation of ultra‐fine dispersions of a second phase.
The effects of several alloying
elements on the yield strength of copper.
Resistance to dislocation motion (loss in ductility)
Microstructure
The structure observed under a microscope
Al Brake – more than one phase
Iron‐chromium alloy – one phase (solid solution)
Phase Equilibria
• Free energy: a function of the internal energy of a system
• Equilibrium: a system is at equilibrium if its free energy is at a minimum
• Phase equilibrium: for a system which has more than one phase
• Phase Diagram is a diagram with T and Composition as axes. They define the stability of the phases that can occur in an alloy system at constant pressure (P). The plots consist of temperature (vertical) axis and compositional (horizontal) axis.
• Constitution: is described by
(a) the phases present
(b) the composition of each phase
(c) the weight fraction of each phase
Isomorphous Phase Diagrams
Binary phase diagram ‐ A phase diagram for a system with two components (C=2).
Ternary phase diagram ‐ A phase diagram for a system with three components (C=3).
Isomorphous phase diagram ‐ A phase diagram in which components display unlimited solid solubility.
Liquidus temperature ‐ The temperature at which the first solid begins to form during solidification.
Solidus temperature ‐ The temperature below which all liquid has completely solidified.
Freezing range – between the liquidus and solidus.
Binary isomorphous systems
• Binary alloy: A mixture of two metals is called a binary alloy and constitute a two‐component system.
• Each metallic element in an alloy is called a separate component.
[Sometimes a compound is considered a component, (e.g., iron carbide)]
• Isomorphous System: In some metallic systems, the two elements are completely soluble in each other in both the liquid and solid states. In these systems only a single type of crystal structure
exists for all compositions of the components (alloy) and therefore it is called isomorphous system.
Example: Binary Isomorphous System (Cu – Ni)
T<1085oC: Cu & Ni are mutually soluble in solid state – complete solubility →
• both have the same FCC structures, • atomic radii and electronegativities are nearly identical
• similar valences
→ isomorphous
Interpretation of Phase Diagrams
Constitution: is described by
(a) the phases present
(b) the composition of each phase
(c) the weight fraction of each phase
(a) Phases Present
Point A: at T=1100oC
60wt% Ni – 40wt% Cu
Only α phase is present
Point B: at T= 1250oC
35wt%Ni – 65wt% Cu Both α & liquid phases are present at equilibrium
(b) Composition of each phase
Single phase:
Point A: 60wt%Ni – 40%Cu alloy at 1100oC
Two‐phase region:
Tie line: across the two‐
phase region at the temperature of the alloy
Point B: T=1250oC
Composition of Liquid phase: CL=31.5wt%Ni – 68.5%Cu
Composition of α phase:
Cα=42.5wt%Ni‐ 57.5wt%Cu
(c) Weight fraction of each phase
Single phase: 100% Ex: Point A: 100% α phase
Two‐phase region: Ex: Point B LEVER RULE (Inverse Lever Rule)
S
WL =
R+S
Cα − Co
WL =
Cα − CL
R
Wα =
R+S
Co − CL
Wα =
Cα − CL
co − c L
35 − 31.5
= 0.32 or 32%
=
Wα =
c s − c L 42.5 − 31.5
Example: Point B: C0 = 35wt%Ni
Cα = 42.5%, CL = 31.5%
c s − co
42.5 − 35
=
= 0.68 or 68%
WL =
c s − c L 42.5 − 31.5
Volume fraction
For an alloy consisting of α and β phases, the volume fraction of the α phase is defined as
vα
Vα =
,
vα + v β
Vα + Vβ = 1
v α ρα
Wα =
;
v α ρα + v β ρ β
V
α
=
Wβ =
W
ρ
W
ρ
α
α
+
Then, the weight fractions are
v β ρβ
Where να and νβ are the volumes of α and β
v α ρα + v β ρ β
α
α
V
W
β
ρ
β
β
=
W
β
ρ
β
W
W α
+
ρα
ρ
β
β
Derivation of the lever rule
1) All material must be in one phase or the other:
Wα + WL = 1
2) Mass of a component that is present in both phases equal to the mass of the component in one phase + mass of the component in the second phase:
Wα cα + WL c L = c o
3) Solution of these equations gives us the Lever rule.
co − cL
Wα =
cα − c L
cα − c o
WL =
cα − c L
Equilibrium Cooling ‐ Development of Microstructure in Isomorphous Alloys
Example: 35wt%Cu‐65wt%Ni system – Slow cooling
from point a to point e
a: 1300oC: complete
liquid with 35wt%Cu65wt%Ni
b: ~1260oC: first solid begin
to form
(α-46wt%Ni)
c: ~1250oC: α-43wt%Ni, L32wt%Ni
d:~1220oC: last liquid to
solidify
e: 35wt%Cu – 65wt%Ni
solid phase
Nonequilibrium Cooling ‐ Development of Microstructure in Isomorphous Alloys
Fast cooling
Compositional changes require diffusion
•Diffusion in the solid state is very slow. ⇒ The new layers that solidify on top of the existing grains have the equilibrium composition at that temperature ⇒ Formation of layered (cored) grains. Tie‐line method to determine the composition of the solid phase is invalid.
•The tie‐line method works for the liquid phase, where diffusion is fast. •Solidus line is shifted to the right (higher Ni contents), solidification is complete at lower T, the outer part of the grains are richer in the low‐melting component (Cu).
•Upon heating grain boundaries will melt first. This can lead to
premature mechanical failure.
Complete solidification occurs at lower temperature and higher Nickel concentration than equilibrium
Solid can’t freeze fast enough: solidus line effectively shifted to higher Ni concentrations. Shift increases with faster cooling rates, slower diffusion
Mechanical properties of isomorphous alloys
Solid solution strengthening
The mechanical properties of copper‐
nickel alloys. Copper is strengthened by up to 60% Ni and nickel is strengthened by up to 40% Cu.
Solidification of a Solid‐Solution Alloy
Segregation ‐ The presence of composition differences in a material, often caused by insufficient time for diffusion during solidification.
Non‐Equilibrium Solidification and Segregation
Coring ‐ Chemical segregation in cast products, also known as microsegregation or interdendritic segregation.
Homogenization heat treatment ‐ The heat treatment used to reduce the microsegregation caused during nonequilibrium solidification.
Macrosegregation ‐ The presence of composition differences in a material over large distances caused by nonequilibrium solidification.
Invariant Points in Binary Systems
•Binary alloys – two components at ambient pressure. Gibbs rule states that P + F = 2 + 1= 3.
•If three phases coexists (P = 3), they coexist at a point (zero degrees of freedom – the invariant point, at a specific temperature and chemical composition
•Types of invariant points:
eutectic, eutectoid, peritectic
peritectoid, monotectic etc.
Five of the Most Important Three Phase Reactions (Invariant Points) in Binary Diagrams
eutectic: Liquid/solid reaction
eutectoid: solid/solid reaction
1150oC: The in‐between point is at 15% B. δ + L are present above the point, γ is present below. The reaction is: δ + L ⇒ γ, a peritectic
920oC: This reaction occurs at 40% B: L1 ⇒ γ + L2 a monotectic
750oC: This reaction occurs at 70% B: L ⇒ γ + β, a eutectic
450oC: This reaction occurs at 20% B: γ ⇒ α + β, a eutectoid
300oC: This reaction occurs at 50% B: α + β ⇒ μ or a peritectoid
Eutectic Systems
• The simplest kind of system with two solid phases is called a eutectic system. • A eutectic system contains two solid phases at low temperature. These phases may have different crystal structures, or the same crystal structure with different lattice parameters. • Examples: – Pb (FCC) and Sn (tetragonal) ‐ solder systems
– Fe (BCC) and C (graphite ‐ hexagonal) ‐ cast irons
– Al (FCC) and Si (diamond cubic) ‐ cast aluminum alloys
– Cu(FCC) and Ag(FCC) – high temperature solder
Cu/Ag Eutectic System
• Copper and Silver are both FCC, but their lattice parameters and
atomic radii are very different, so they have limited solubility in the solid state. • There are two solid stable phases α and β, and at high temperatures there is a eutectic reaction where the solids α, β
and the liquid coexist.
←⎯⎯⎯
L(CE )
α (CαE ) + β (CβE )
⎯Cooling
⎯⎯→
Heating
Hypoeutectic alloy ‐ An alloy composition between that of the left‐
hand‐side end of the tie line defining the eutectic reaction and the eutectic composition.
Hypereutectic alloys ‐ An alloy composition between that of the right‐
hand‐side end of the tie line defining the eutectic reaction and the eutectic composition.
Cu – Ag System
Cu: α phase
Ag: β phase Eutectic means “easily melted” in Greek
Point E: invariant point (eutectic point)
BG line: isotherm line
AB & FG: Solidus line
BC & GH: Solvus line
AE & EF: Liquidus line
BEG: Solidus line, isotherm line
TE: eutectic isotherm
Eutectic isotherm
Invariant or eutectic point
L(CE )
Eutectic Reaction:
For copper‐silver system:
L(71.9 wt % Ag )
Heating
←⎯
⎯
⎯⎯
⎯Cooling
⎯⎯
⎯→
Heating
←⎯
⎯
⎯⎯
⎯Cooling
⎯⎯
⎯→
α (CαE ) + β (CβE )
α (8.0 wt % Ag ) + β (91.2wt % Ag )
Eutectic or invariant point ‐ Liquid and two solid phases co‐exist in equilibrium at the eutectic composition CE and the eutectic temperature TE. Eutectic isotherm ‐ horizontal solidus line at TE.
Binary Eutectic System
Eutectic reaction – transition from liquid to mixture of two solid phases, α + β at eutectic concentration CE. At most two phases can be in equilibrium. Three phases (L, α, β) may be in equilibrium only at a few points along the eutectic isotherm. Single‐phase regions are separated by 2‐
phase regions.
Binary Eutectic System
Compositions and relative amounts of phases are determined from the same tie lines and lever rule, as for isomorphous alloys‐‐
demonstrate
•A
•B
•C
Example
For Point C: 40wt%Sn‐60wt%Pb alloy at 150oC
a) What are the phases present?
b) What are the compositions of the phases present?
c) Mass fraction?
d) Volume fraction at 150oC?
Knowing that the densities of Pb and Sn are 11.23 and 7.24g/cm3, respectively a) At C, α and β phases coexist
b) Draw Tie Line at 150oC: For α‐phase:
Cα = 10% →10wt%Sn–90wt%Pb
For β‐phase:
Cβ = 98%
→98wt%Sn–2wt%Pb
•C
Cβ − C1 98 − 40
=
= 0.66
Wα =
Cβ − Cα 98 − 10
c) Mass fraction:
Wβ =
C1 − Cα 40 − 10
=
= 0.34
Cβ − Cα 98 − 10
Wα
0.66
να
ρα
10.64
=
=
= 0.57
d) volume Vα =
W
0
.
66
0
.
34
ν α + ν β Wα
+
+ β
fraction:
10
.
64
7.39
ρα
ρβ
Vβ = 1 − Vα = 1 − 0.57 = 0.43
where
ρα =
100
=
C Sn −α C Pb −α
+
ρ Sn
ρα =
ρ Pb
100
=
C Sn − β C Pb − β
+
ρ Sn
ρ Pb
100
10
90
+
7.24 g .cm − 3 11 .23 g .cm − 3
100
98
2
+
7.24 g .cm − 3 11 .23 g .cm − 3
= 10 .64 g .cm − 3
= 7.29 g .cm − 3
L
⇓
L+α
⇓
α
Development of microstructure in
eutectic alloys (I)
Several types of microstructure
formed in slow cooling an different
compositions.
Cooling of liquid lead/tin system at
different compositions.
In this case of lead-rich alloy (0-2
wt% of tin) solidification proceeds in
the same manner as for
isomorphous alloys (e.g. Cu-Ni) that
we discussed earlier.
Development of microstructure in eutectic alloys (II)
L
α +L At compositions between room temperature solubility α
limit and the maximum solid solubility at the eutectic α +β temperature, β phase nucleates as the α solid solubility is exceeded at solvus line.
Development of microstructure in eutectic alloys (III)
Solidification at the eutectic composition (I)
No changes above eutectic temperature, TE. At TE liquid transforms to α and β phases (eutectic reaction).
L → α +β
Development of microstructure in eutectic alloys (IV)
Solidification at the eutectic composition (II)
Compositions of α and β phases are very different → eutectic reaction involves redistribution of Pb and Sn atoms by atomic diffusion. Simultaneous formation of α and β phases results in a layered (lamellar) microstructure:called eutectic structure.
Formation of eutectic structure in lead‐tin system.
Dark layers are lead‐
reach α phase.
Light layers are the tin‐reach β phase. Development of microstructure in eutectic alloys (V) Compositions other than eutectic but within the range of the eutectic isotherm
Primary α phase is formed in the α + L region, and the eutectic structure that includes layers of α and β phases (called eutectic α and eutectic β phases) is formed upon crossing the eutectic isotherm. L
⇓
α+L
⇓
α+β
Development of microstructure in eutectic alloys (VI)
Microconstituent – element of microstructure having a distinctive structure. For case described on previous page, microstructure consists of two microconstituents, primary α
phase and the eutectic structure.
Although the eutectic structure consists of two phases, it is a microconstituent with distinct lamellar structure and fixed ratio of the two phases.
Compositions of α and β phases are very different → eutectic reaction involves redistribution of Pb and Sn atoms by atomic diffusion. Simultaneous formation of α and β phases results in a layered (lamellar) microstructure:called eutectic structure.
Formation of eutectic structure in lead‐tin system.
Dark layers are lead‐
reach α phase.
Light layers are the tin‐
reach β phase.
Relative amounts of microconstituents?
Eutectic microconstituent forms from liquid having eutectic composition (61.9 wt% Sn)
P
We =
....eutectic
(P + Q )
Treat the eutectic as if it were a separate phase and apply lever rule to find relative Q
Wα =
.... primary
fractions of primary α phase (18.3wt% Sn) and (P + Q )
eutectic structure (61.9wt% Sn):
Terminal solid solution: a solid solution that exists over a composition range extending to either composition extremity of a binary phase diagram.
(a) A hypoeutectic lead‐tin alloy. (b) A hypereutectic lead‐tin alloy. The dark constituent is the lead‐rich solid α, the light
constituent is the tin‐rich solid β, and the fine plate structure is the eutectic (x400).
The effect of the chemical composition and strengthening mechanism on the tensile strength of lead‐tin alloys.
Eutectic colonies and interlamellar spacing
Equilibrium Diagrams Having Intermediate Phases or Compounds
Intermediate solid solution:
Copper-zinc
α and η: two terminal solid solution
δ
β
α
β, γ, δ, & ε are γ
intermediate
phases
ε
Intermetallic Compounds
Ex: magnesium‐
lead phase diagram:
Intermetallic
compound: Mg2Pb can exist by itself only at the precise composition of 19wt%Mg –
81wt%Pb
Eutectoid Reaction (Invariant Point E at 560oC)
Copper‐zinc δ
cooling
heating
γ +ε
Eutectoid reaction
Peritectic Reaction (Invariant Point P at 598oC)
Copper‐zinc δ+L
cooling
heating
ε
Peritectic reaction