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Material Type: Notes; Professor: Zhigilei; Class: Thermodynamics and Kinetics of Materials; Subject: Materials Science and Engineering; University: University of Virginia; Term: Unknown 1989;
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MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Binary phase diagrams
Binary phase diagrams and Gibbs free energy curves
Binary solutions with unlimited solubility
Relative proportion of phases (tie lines and the lever principle)
Development of microstructure in isomorphous alloys
Binary eutectic systems (limited solid solubility)
Solid state reactions (eutectoid, peritectoid reactions)
Binary systems with intermediate phases/compounds
The iron-carbon system (steel and cast iron)
Gibbs phase rule
Temperature dependence of solubility
Three-component (ternary) phase diagrams
Reading: Chapters 1.5.1 – 1.5.7 of Porter and Easterling, Chapter 10 of Gaskell
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Binary phase diagram and Gibbs free energy
α
A binary phase diagram is a temperature - composition map which indicates the equilibrium phases present at a given temperature and composition.
The equilibrium state can be found from the Gibbs free energy dependence on temperature and composition.
α
We have also discussed the dependence of the Gibbs free energy from composition at a given T:
We have discussed the dependence of G of a one- component system on T:
S T
G P
⎟=^ − ⎠
⎞ ⎜ ⎝
⎛ ∂
∂
T
c T
S T
G (^) P P P
2
2 ⎟=^ − ⎠
⎜ ⎞ ⎝
⎛ ∂
=−∂ ⎟⎟ ⎠
⎞ ⎜⎜ ⎝
⎛ ∂
∂
G =XA GA+XBGB+ΔHmix−TΔS mix
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Binary solutions with unlimited solubility
liquid
Let’s construct a binary phase diagram for the simplest case: A and B components are mutually soluble in any amounts in both solid ( isomorphous system ) and liquid phases, and form ideal solutions.
We have 2 phases – liquid and solid. Let’s consider Gibbs free energy curves for the two phases at different T
liquid
pure components: T 1 > T (^) m(A) > T (^) m(B) → the liquid phase will be the stable phase for any composition.
solid
solid
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Binary solutions with unlimited solubility (II)
solid
Decreasing the temperature below T 1 will have two effects:
will increase more rapidly than
liquid
component A, where
liquid B
liquid
solid
solid
solid A
liquid
solid A
liquid
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Binary solutions with unlimited solubility (III)
solid
energy curves for the liquid and solid phases will cross.
liquid
T 3
solid
As we discussed before, the common tangent construction can be used to show that for compositions near cross-over of Gsolid^ and Gliquid^ , the total Gibbs free energy can be minimized by separation into two phases.
0 XB 1
solid liquid
solid + liquid X 1 X 2
liquid
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Binary solutions with unlimited solubility (IV)
liquid
solid B
liquid
T 4
solid
At T 4 and below this temperature the Gibbs free energy of the solid phase is lower than the G of the liquid phase in the whole range of compositions – the solid phase is the only stable phase.
0 XB^1
As temperature decreases below T 3 continue
to increase more rapidly than
¾ Therefore, the intersection of the Gibbs free energy curves, as
= T (^) m(B) the curves will intersect at X 1 = X 2 = 1
liquid B
liquid
solid B
solid
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Binary solutions with unlimited solubility (V)
solid
Based on the Gibbs free energy curves we can now construct a phase diagram for a binary isomorphous systems
liquid
T 3 solid
0 XB 1
solid solid + liquid liquid
liquid
T 2
T 3
T 4
T 5
T T^1
T 4
T 2
T 1
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Liquidus line separates liquid from liquid + solid Solidus line separates solid from liquid + solid
Binary solutions with unlimited solubility (VI) Example of isomorphous system: Cu-Ni (the complete solubility occurs because both Cu and Ni have the same crystal structure, FCC, similar radii, electronegativity and valence).
Liquid
α
Solid solution
Liquidus line Solidus line
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Composition Conversions
Weight % to Atomic %:
Atomic % to Weight %:
B
wt A A
wt B
A
wt at B
B
wt A A
wt B
B
wt at A
W (^) L = (C wtα - C wto ) / (C wtα - C wtL)
Of course the lever rule can be formulated for any specification of composition:
ML^ = (X (^) Bα^ - X (^) B^0 )/(X (^) Bα^ - X (^) BL) = (Catα - Cato) / (Catα - CatL)
Mα^ = (X (^) B^0 - X (^) BL)/(X (^) Bα^ - X (^) BL) = (Cat 0 - CatL) / (Catα - CatL)
Wα = (C wto - C wtL) / (C wtα - C wtL)
A
at B A
at B
B
at wt B B ×
A
at B A
at B
A
at wt A A ×
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Mass fractions: W (^) L = S / (R+S) = (Cα - Co ) / (Cα - CL) = 0.
Wα = R / (R+S) = (Co - CL) / (Cα - C (^) L) = 0.
Co = 35 wt. %, CL = 31.5 wt. %, Cα = 42.5 wt. %
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Development of microstructure in isomorphous alloys Equilibrium (very slow) cooling
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Development of microstructure in isomorphous alloys Equilibrium (very slow) cooling
¾ Solidification in the solid + liquid phase occurs gradually upon cooling from the liquidus line.
¾ The composition of the solid and the liquid change gradually during cooling (as can be determined by the tie-line method.)
¾ Nuclei of the solid phase form and they grow to consume all the liquid at the solidus line.
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Development of microstructure in isomorphous alloys Non-equilibrium cooling
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Development of microstructure in isomorphous alloys Non-equilibrium cooling
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Binary solutions with a miscibility gap Let’s consider a system in which the liquid phase is approximately ideal, but for the solid phase we have Δ H (^) mix > 0
0 XB^1
T 2 <T (^1) Gliquid
0 X B^1
0 X B^1
T
0 XB^1
T 3
T 1
T 2
liquid
α
α 1 +α 2
At low temperatures, there is a region where the solid solution is most stable as a mixture of two phases α 1 and α 2 with compositions X 1 and X 2. This region is called a miscibility gap.
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Eutectic phase diagram
For an even larger ΔH (^) mix the miscibility gap can extend into the liquid phase region. In this case we have eutectic phase diagram.
0 XB 1
0 X B 1
T
0 XB^1
T 3
T 1 T 2
liquid
α 1 α 1 +α 2
0 X B^1
α 2
α 1 +l α 2 +l
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Compositions and relative amounts of phases are determined from the same tie lines and lever rule, as for isomorphous alloys
For points A, B, and C calculate the compositions (wt. %) and relative amounts (mass fractions) of phases present.
Composition, wt% Sn
Temperature,
°
C
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Development of microstructure in eutectic alloys (I)
Several different types of microstructure can be formed in slow cooling an different compositions. Let’s consider cooling of liquid lead – tin system as an example.
In the 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.
L → α +L → α
Composition, wt% Sn
Temperature,
°C
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Development of microstructure in eutectic alloys (II)
At compositions between the room temperature solubility limit and the maximum solid solubility at the eutectic temperature, β phase nucleates as the α solid solubility is exceeded upon crossing the solvus line.
L
α +L
α
α + β
Composition, wt% Sn
Temperature,
°
C
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
No changes above the eutectic temperature TE. At TE all the liquid transforms to α and β phases ( eutectic reaction ).
L → α + β
Development of microstructure in eutectic alloys (III) Solidification at the eutectic composition
Composition, wt% Sn
Temperature,
°
C
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Development of microstructure in eutectic alloys (IV) Solidification at the eutectic composition
Compositions of α and β phases are very different → eutectic reaction involves redistribution of Pb and Sn atoms by atomic diffusion (we will learn about diffusion in the last part of this course). This simultaneous formation of α and β phases result in a layered (lamellar) microstructure that is called eutectic structure.
Formation of the eutectic structure in the lead-tin system. In the micrograph, the dark layers are lead-reach α phase, the light layers are the tin-reach β phase.
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
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.
Composition, wt% Sn
Temperature,
°C
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Development of microstructure in eutectic alloys (VI)
Microconstituent – element of the microstructure having a distinctive structure. In the case described in the 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.
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
How to calculate relative amounts of microconstituents?
Eutectic microconstituent forms from liquid having eutectic composition (61.9 wt% Sn) We can treat the eutectic as a separate phase and apply the lever rule to find the relative fractions of primary α phase (18.3 wt% Sn) and the eutectic structure (61.9 wt% Sn):
We = P / (P+Q) (eutectic) W α ’ = Q / (P+Q) (primary)
Composition, wt% Sn
Temperature,
°C
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Phase diagrams with intermediate phases: example
Example of intermediate solid solution phases: in Cu-Zn, α and η are terminal solid solutions, β, β’, γ, δ, ε are intermediate solid solutions.
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Phase diagrams for systems containing compounds For some systems, instead of an intermediate phase, an intermetallic compound of specific composition forms. Compound is represented on the phase diagram as a vertical line, since the composition is a specific value. When using the lever rule, compound is treated like any other phase, except they appear not as a wide region but as a vertical line
This diagram can be thought of as two joined eutectic diagrams, for Mg-Mg 2 Pb and Mg 2 Pb-Pb. In this case compound Mg 2 Pb (19 %wt Mg and 81 %wt Pb) can be considered as a component. A sharp drop in the Gibbs free energy at the compound composition should be added to Gibbs free energy curves for the existing phases in the system.
intermetallic compound
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Stoichiometric and non-stoichiometric compounds Compounds which have a single well-defined composition are called stoichiometric compounds (typically denoted by their chemical formula). Compound with composition that can vary over a finite range are called non-stoichiometric compounds or intermediate phase (typically denoted by Greek letters).
0 XB^1
liquid
α
γ (^) β + γ
β
0 XB^1
liquid
α α + AB β
AB^ β^ +^ AB
l +AB
β
non-stoichiometric stoichiometric
Common stoichiometric compounds:
composition, at% 50.0 45.5 44.4 42.9 40.0 37.
compound AB A 6 B 5 A 5 B 4 A 4 B 3 A 3 B 2 A 5 B 3
composition, at% 33.3 28.6 25.0 20.0 16.7 14.
compound A 2 B A 5 B 2 A 3 B A 4 B A 5 B A 6 B
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Eutectoid Reactions
The eutectoid ( eutectic-like in Greek) reaction is similar to the eutectic reaction but occurs from one solid phase to two new solid phases.
Eutectoid structures are similar to eutectic structures but are much finer in scale (diffusion is much slower in the solid state).
Upon cooling, a solid phase transforms into two other solid phases (δ ↔ γ + ε in the example below)
Looks as V on top of a horizontal tie line (eutectoid isotherm) in the phase diagram.
Eutectoid
Cu-Zn
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Eutectic and Eutectoid Reactions
The above phase diagram contains both an eutectic reaction and its solid-state analog, an eutectoid reaction
α
α +β
γ+ l
α+γ
l +β γ
l
Temperature
Eutectic temperature
Eutectoid temperature
Eutectoid composition
Eutectic composition
Composition
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Peritectic Reactions
A peritectic reaction - solid phase and liquid phase will together form a second solid phase at a particular temperature and composition upon cooling - e.g. L + α ↔ β
These reactions are rather slow as the product phase will form at the boundary between the two reacting phases thus separating them, and slowing down any further reaction.
Peritectoid is a three-phase reaction similar to peritectic but occurs from two solid phases to one new solid phase (α + β = γ).
Temperature
α+ liquid
β (^) β+ liquid
liquid
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Example: The Iron–Iron Carbide (Fe–Fe 3 C) Phase Diagram
In their simplest form, steels are alloys of Iron (Fe) and Carbon (C). The Fe-C phase diagram is a fairly complex one, but we will only consider the steel part of the diagram, up to around 7% Carbon.
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Phases in Fe–Fe 3 C Phase Diagram
¾ α -ferrite - solid solution of C in BCC Fe
¾ γ -austenite - solid solution of C in FCC Fe
¾ δ -ferrite solid solution of C in BCC Fe
¾ Fe 3 C (iron carbide or cementite)
¾ Fe-C liquid solution
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Compositions to the left of eutectoid (0.022 - 0.76 wt % C) hypoeutectoid ( less than eutectoid -Greek) alloys.
Microstructure of hypoeutectoid steel (I)
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Hypoeutectoid alloys contain proeutectoid ferrite (formed above the eutectoid temperature) plus the eutectoid perlite that contain eutectoid ferrite and cementite.
Microstructure of hypoeutectoid steel (II)
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Compositions to the right of eutectoid (0.76 - 2.14 wt % C) hypereutectoid ( more than eutectoid -Greek) alloys.
Microstructure of hypereutectoid steel (I)
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Microstructure of hypereutectoid steel
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Hypereutectoid alloys contain proeutectoid cementite (formed above the eutectoid temperature) plus perlite that contain eutectoid ferrite and cementite.
Microstructure of hypereutectoid steel (II)
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
How to calculate the relative amounts of proeutectoid phase (α or Fe 3 C) and pearlite?
Application of the lever rule with tie line that extends from the eutectoid composition (0.75 wt% C) to α – (α + Fe 3 C) boundary (0.022 wt% C) for hypoeutectoid alloys and to (α + Fe 3 C) – Fe 3 C boundary (6.7 wt% C) for hipereutectoid alloys.
Fraction of α phase is determined by application of the lever rule across the entire (α + Fe 3 C) phase field:
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Example for hypereutectoid alloy with composition C 1
Fraction of pearlite:
WP = X / (V+X) = (6.7 – C 1 ) / (6.7 – 0.76)
Fraction of proeutectoid cementite:
WFe3C = V / (V+X) = (C 1 – 0.76) / (6.7 – 0.76)
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
The Most Important Phase Diagram in the History of Civilization
naturally formed composite: hard & brittle ceramic (Fe 3 C)
soft BCC iron (ferrite)
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Multicomponent systems (II)
The Gibbs free energy surfaces (instead of curves for a binary system) can be plotted for all the possible phases and for different temperatures.
The chemical potentials of A, B, and C of any phase in this system are given by the points where the tangential plane to the free energy surfaces intersects the A, B, and C axis.
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Multicomponent systems (III) A three-phase equilibrium in the ternary system for a given temperature can be derived by means of the tangential plane construction.
Eutectic point: four-phase equilibrium between α, β, γ, and liquid
α β
γ
For two phases to be in equilibrium, the chemical potentials should be equal, that is the compositions of the two phases in equilibrium must be given by points connected by a common tangential plane (e.g. l and m). The relative amounts of phases are given by the lever rule (e.g. using tie-line l-m). A three phase triangle can result from a common tangential plane simultaneously touching the Gibbs free energies of three phases (e.g. points x, y, and z).
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
An example of ternary system
The ternary diagram of Ni-Cr-Fe. It includes Stainless Steel (wt.% of Cr > 11.5 %, wt.% of Fe > 50 %) and Inconeltm^ (Nickel based super alloys). Inconel have very good corrosion resistance, but are more expensive and therefore used in corrosive environments where Stainless Steels are not sufficient (Piping on Nuclear Reactors or Steam Generators).
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Another example of ternary phase diagram: oil – water – surfactant system
Surfactants are surface-active molecules that can form interfaces between immiscible fluids (such as oil and water). A large number of structurally different phases can be formed, such as droplet, rod-like, and bicontinuous microemulsions, along with hexagonal, lamellar, and cubic liquid crystalline phases.
Drawing by Carlos Co, University of Cincinnati
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Summary Elements of phase diagrams:
α + β
α β
α + β
α β
γ
α + L α β
Eutectic (L → α + β)
Eutectoid (γ → α + β)
Peritectic (α + L → β)
α + β α γ
β
Peritectoid (α + β → γ)
Compound, AnB (^) m
Make sure you understand language and concepts:
¾ Common tangent construction ¾ Separation into 2 phases ¾ Eutectic structure ¾ Composition of phases
¾ Weight and atom percent ¾ Miscibility gap ¾ Solubility dependence on T ¾ Intermediate solid solution ¾ Compound ¾ Isomorphous ¾ Tie line, Lever rule ¾ Liquidus & Solidus lines ¾ Microconstituent
¾ Primary phase ¾ Solvus line, Solubility limit ¾ Austenite, Cementite, Ferrite ¾ Pearlite ¾ Hypereutectoid alloy ¾ Hypoeutectoid alloy ¾ Ternary alloys ¾ Gibbs phase rule
A useful link: http://www.soton.ac.uk/~pasr1/index.htm