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 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Binary phase diagram and Gibbs free energy
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.
G
We have discussed the
dependence of G of a onecomponent system on T:
α
⎛ ∂G ⎞
⎜
⎟ = −S
T
∂
⎝
⎠P
β
⎛ ∂ 2G ⎞
c
⎛ ∂S ⎞
⎜⎜ 2 ⎟⎟ = −⎜ ⎟ = − P
T
⎝ ∂T ⎠ P
⎝ ∂T ⎠ P
We have also discussed the
dependence of the Gibbs free
energy from composition at a
given T:
T
GαA
GαB
G = XAGA + XBGB + ΔHmix − TΔ Smix
Gα
0
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
XB
1
Binary solutions with unlimited solubility
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
¾ T1 is above the equilibrium melting temperatures of both
pure components: T1 > Tm(A) > Tm(B) → the liquid phase
will be the stable phase for any composition.
G solid
B
G
T1
solid
A
G liquid
A
G liquid
B
G solid
G liquid
0
XB
1
G id = X A G A + X BG B + RT[X A lnXA + X BlnXB ]
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Binary solutions with unlimited solubility (II)
Decreasing the temperature below T1 will have two effects:
liquid
™ GA
solid
and G liquid
G
will
increase
more
rapidly
than
B
A
and G solid
B
Why?
™ The curvature of the G(XB) curves will decrease.
¾ Eventually we will reach
T2 – melting point of pure
solid
component A, where G liquid
=
G
A
A
G
liquid
A
=G
G solid
B
T2
solid
A
G liquid
B
G solid
G liquid
0
Why?
XB
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
1
Binary solutions with unlimited solubility (III)
¾ For even lower temperature T3 < T2 = Tm(A) the Gibbs free
energy curves for the liquid and solid phases will cross.
T3
G liquid
A
G solid
B
G solid
G solid
A
G liquid
B
G liquid
solid
0
X1
solid +
liquid
XB
liquid
X2
1
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.
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Binary solutions with unlimited solubility (IV)
liquid
continue
Gliquid
and
G
A
B
solid
solid
to increase more rapidly than GA and GB
As temperature decreases below T3
¾ Therefore, the intersection of the Gibbs free energy curves, as
well as points X1 and X2 are shifting to the right, until, at T4
= Tm(B) the curves will intersect at X1 = X2 = 1
T4
G liquid
A
G liquid
solid
G liquid
=
G
B
B
G solid
A
G solid
0
XB
1
At T4 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.
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Binary solutions with unlimited solubility (V)
Based on the Gibbs free energy curves we can now construct a
phase diagram for a binary isomorphous systems
G
liquid
A
T3
G solid
A
G solid
B
G solid
G liquid
B
G liquid
T1
solid
s
solid +
liquid
liquid
l
s
T2
T1
T
T2
l
l
s
T3
T4
T4
T5
0
XB
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
1
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
Liquidus line
Solidus line
α
Solid solution
Liquidus line separates liquid from liquid + solid
MSE
3050, line
Phaseseparates
Diagrams and
Kinetics,
Leonid
Zhigilei
Solidus
solid
from
liquid
+ solid
Binary solutions with unlimited solubility (VII)
In one-component system melting occurs at a well-defined
melting temperature.
In multi-component systems melting occurs over the range of
temperatures, between the solidus and liquidus lines. Solid and
liquid phases are at equilibrium in this temperature range.
Liquidus
Temperature
L
liquid solution
α+L
Solidus
liquid solution
+
crystallites of
solid solution
α
polycrystal
solid solution
A
20 40 60 80
Composition, wt %
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
B
Interpretation of Phase Diagrams
For a given temperature and composition we can use phase
diagram to determine:
1) The phases that are present
2) Compositions of the phases
3) The relative fractions of the phases
Finding the composition in a two phase region:
1.
Locate composition and temperature in diagram
2.
In two phase region draw the tie line or isotherm
3.
Note intersection with phase boundaries. Read compositions
at the intersections.
The liquid and solid phases have these compositions.
XB
Xliquid
B
Xsolid
B
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Interpretation of Phase Diagrams: the Lever Rule
Finding the amounts of phases in a two phase region:
1.
Locate composition and temperature in diagram
2.
In two phase region draw the tie line or isotherm
3.
Fraction of a phase is determined by taking the length of the
tie line to the phase boundary for the other phase, and
dividing by the total length of tie line
α+β
α
β
The lever rule is a mechanical
analogy to the mass balance
calculation. The tie line in the twophase region is analogous to a lever
balanced on a fulcrum.
Wα
Wβ
Derivation of the lever rule:
1) All material must be in one phase or the other: Wα + Wβ = 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: WαCα + WβCβ = Co
3) Solution of these equations gives us the Lever rule.
Wβ = (C0 - Cα) / (Cβ - Cα) and Wα = (Cβ- C0) / (Cβ - Cα)
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Composition/Concentration:
weight fraction vs. molar fraction
Composition can be expressed in
Molar fraction, XB, or atom percent (at %) that is useful when
trying to understand the material at the atomic level. Atom
percent (at %) is a number of moles (atoms) of a particular
element relative to the total number of moles (atoms) in alloy.
For two-component system, concentration of element B in at. %
is
nB
C at =
m
n Bm + n Am
× 100
C at = X B × 100
Where nmA and nmB are numbers of moles of elements A and B in
the system.
Weight percent (C, wt %) that is useful when making the
solution. Weight percent is the weight of a particular component
relative to the total alloy weight. For two-component system,
concentration of element B in wt. % is
C wt =
mB
× 100
mB + mA
where mA and mB are the weights of the components in the
system.
n Am =
mA
AA
n Bm =
mB
AB
where AA and AB are atomic
weights of elements A and B.
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Composition Conversions
Weight % to Atomic %:
C atB
C Bwt A A
= wt
× 100
wt
CB AA + CA AB
wt
C
at
A AB
C A = wt
× 100
wt
CB AA + CA A B
Atomic % to Weight %:
C Bwt
C atB A B
= at
× 100
at
CBA B + CAA A
wt
A
C atA A A
= at
× 100
at
CBA B + CAA A
C
Of course the lever rule can be formulated for any specification
of composition:
ML = (XBα - XB0)/(XBα - XBL) = (Catα - Cato) / (Catα - CatL)
Mα = (XB0 - XBL)/(XBα - XBL) = (Cat0 - CatL) / (Catα - CatL)
WL = (Cwtα - Cwto) / (Cwtα - CwtL)
Wα = (Cwto- CwtL) / (Cwtα - CwtL)
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Phase compositions and amounts. An example.
Co = 35 wt. %, CL = 31.5 wt. %, Cα = 42.5 wt. %
Mass fractions: WL = S / (R+S) = (Cα - Co) / (Cα - CL) = 0.68
Wα = R / (R+S) = (Co - CL) / (Cα - CL) = 0.32
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Development of microstructure in isomorphous alloys
Equilibrium (very slow) cooling
MSE 3050, 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 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Development of microstructure in isomorphous alloys
Non-equilibrium cooling
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Development of microstructure in isomorphous alloys
Non-equilibrium cooling
•
Compositional changes require diffusion in solid and liquid
phases
•
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 but once they are solid their
composition does not change. ⇒ Formation of layered grains
and the invalidity of the tie-line method to determine the
composition of the solid phase.
•
The tie-line method still works for the liquid phase, where
diffusion is fast. Average Ni content of solid grains is higher.
⇒ Application of the lever rule gives us a greater proportion
of liquid phase as compared to the one for equilibrium
cooling at the same T. ⇒ 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.
MSE 3050, 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 ΔHmix > 0
T1
G
T2 < T1
G
G liquid
G solid
G solid
G liquid
0
1
XB
G
liquid
XB
T
T1
T3 < T2
G
0
liquid
T2
G solid
α
T3
0
1
XB
1
α1+α2
0
XB
1
At low temperatures, there is a region where the solid solution is
most stable as a mixture of two phases α1 and α2 with
compositions X1 and X2. This region is called a miscibility gap.
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
T
G
liquid
T1
T2
0
XB
1
0
XB
1
α
0
G
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
α1+α2
XB
1
G reg = X A G A + X BG B + ΩX A X B + RT[X A lnXA + X BlnXB ]
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Eutectic phase diagram
For an even larger ΔHmix the miscibility gap can extend into the
liquid phase region. In this case we have eutectic phase
diagram.
G
G
G
T1
liquid
G liquid
G solid
0
T3 < T2
G
G
G solid
1
XB
liquid
XB
0
T
T1
T2
T3
G solid
0
T2 < T1
1
1
XB
liquid
α1+l
α1
0
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
α2+l
α1+α2
XB
α2
1
Eutectic phase diagram with different crystal
structures of pure phases
A similar eutectic phase diagram can result if pure A and B have
different crystal structures.
T1
G
T2 < T1
G
α
β
α
β
liquid
liquid
0
1
XB
T2
liquid
0
T3
XB
liquid
T1
α
β
1
XB
T
T3 < T2
G
0
1
α+l
β+l
α
β
α+β
0
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
XB
1
Temperature dependence of solubility
There is limited solid solubility of A in B and B in A in the alloy
having eutectic phase diagram shown below.
G
T1
β
T
liquid
α
α+l
liquid
0
T1
XB
1
β+l
α
β
α+β
0
XB
1
The closer is the minimum of the Gibbs free energy curve
Gα(XB) to the axes XB = 0, the smaller is the maximum possible
concentration of B in phase α. Therefore, to discuss the
temperature dependence of solubility let’s find the minimum of
Gα(XB).
G reg = X A G A + X BG B + ΩX A X B + RT[X A lnX A + X BlnX B ]
dG
=0
dX B
- Minimum of G(XB)
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Temperature dependence of solubility (II)
G reg = X A G A + X BG B + ΩX A X B + RT[X A lnX A + X BlnX B ]
= (1 - X B )G A + X BG B + Ω(1 - X B )X B + RT[(1 - X B )ln(1 - X B ) + X BlnXB ]
⎡
(1 - X B ) + lnX + X B ⎤ =
dG
= -G A + G B + Ω − 2Ω X B + RT ⎢- ln(1 - X B ) −
B
⎥
(1 - X B )
dX B
XB ⎦
⎣
= -GA + GB + Ω − 2Ω XB + RT[- ln(1- XB ) + lnXB ] =
⎛ X ⎞
= G B - G A + Ω(1 − 2XB ) + RTln⎜⎜ B ⎟⎟ ≈ GB - G A + Ω + RTln(XB ) = 0
⎝ 1 - XB ⎠
if XB is small (XB → 0)
(GB >> GA)
⎛ G − GA + Ω ⎞
XB ≈ exp⎜ − B
⎟
RT
⎝
⎠
Solid solubility of B in α increases
exponentially with temperature
(similar to vacancy concentration)
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Minimum of G(XB)
Temperature dependence of solubility (III)
α+l
liquid
β+l
α
β
T1
α+β
0
1
XB
⎛ G − GA + Ω ⎞
XB ≈ exp⎜ − B
⎟
RT
⎝
⎠
XB
XB vs. T
T
T vs. XB
T
Te
XB
supersaturation with B
→ phase separation α and β
→ precipitation of B-rich intermediate
phase or compound
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Eutectic systems - alloys with limited solubility (I)
Liquidus
liquid
Temperature, °C
Solidus
Solvus
α+β
Copper – Silver phase diagram
Composition, wt% Ag
Three single phase regions (α - solid solution of Ag in Cu matrix,
β = solid solution of Cu in Ag matrix, L - liquid)
Three two-phase regions (α + L, β +L, α +β)
Solvus line separates one solid solution from a mixture of solid
solutions. Solvus line shows limit of solubility
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Eutectic systems - alloys with limited solubility (II)
Temperature, °C
Lead – Tin phase diagram
Invariant or eutectic point
Eutectic isotherm
Composition, wt% Sn
Eutectic or invariant point - Liquid and two solid phases coexist in equilibrium at the eutectic composition CE and the
eutectic temperature TE.
Eutectic isotherm - the horizontal solidus line at TE.
Eutectic reaction – transition between liquid and mixture of two
solid phases, α + β at eutectic concentration CE.
The melting point of the eutectic alloy is lower than that of the
components (eutectic = easy to melt in Greek).
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Eutectic systems - alloys with limited solubility (III)
Compositions and relative amounts of phases are determined
from the same tie lines and lever rule, as for isomorphous alloys
Temperature, °C
•A
•B
•C
Composition, wt% Sn
For points A, B, and C calculate the compositions (wt. %) and
relative amounts (mass fractions) of phases present.
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Temperature, °C
•C
Composition, wt% Sn
MSE 3050, 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.
Temperature, °C
In the case of lead-rich alloy
(0-2
wt.
%
of
tin)
solidification proceeds in the
same
manner
as
for
isomorphous alloys (e.g. CuNi) that we discussed earlier.
L → α +L→ α
Composition, wt% Sn
MSE 3050, 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.
Temperature, °C
L
α +L
α
α +β
Composition, wt% Sn
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Development of microstructure in eutectic alloys (III)
Solidification at the eutectic composition
Temperature, °C
No changes above the eutectic temperature TE. At TE all the
liquid transforms to α and β phases (eutectic reaction).
Composition, wt% Sn
L → α +β
MSE 3050, 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 3050, 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.
Temperature, °C
L → α + L → α +β
Composition, wt% Sn
MSE 3050, 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 3050, 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):
Wα’ = Q / (P+Q) (primary)
Temperature, °C
We = P / (P+Q) (eutectic)
Composition, wt% Sn
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Temperature, °C
Composition, wt% Sn
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
How to calculate the total amount of α phase (both eutectic
and primary)?
Fraction of α phase determined by application of the lever rule
across the entire α + β phase field:
Wα = (Q+R) / (P+Q+R) (α phase)
Temperature, °C
Wβ = P / (P+Q+R) (β phase)
Composition, wt% Sn
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
T
liquid
0
XB
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
1
Monotectic system
T
α + L1
Upper solidus
L1
Monotectic point
Lower solidus
Limit of liquid
immiscibility
Lower
liquidus
L1 + L2
α
L2
α + L2
β+L2
Monotectic isotherm
β
α+β
0
F.N. Rhines, Phase Diagrams in Metallurgy, 1956
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
XB
1
Binary solutions with ΔHmix < 0 - ordering
If ΔHmix < 0 bonding becomes stronger upon mixing → melting
point of the mixture will be higher than the ones of the pure
components. For the solid phase strong interaction between
unlike atoms can lead to (partial) ordering → |ΔHmix| can
become larger than |ΩXAXB| and the Gibbs free energy curve for
the solid phase can become steeper than the one for liquid.
G
G
T1
T2 < T1
Gα
G liquid
G liquid
0
G G
Gα
1
XB
liquid
0
T
T1
T3 < T2
1
XB
liquid
T2
α
Gα
T3
G
0
XB
α`
1
α`
0
XB
At low temperatures, strong attraction between
unlike atoms can lead to the formation of
MSE
3050, Phase
Diagrams
ordered
phase
α`. and Kinetics, Leonid Zhigilei
1
Binary solutions with ΔHmix < 0 - intermediate phases
If attraction between unlike atoms is very strong, the ordered
phase may extend up to the liquid.
T
liquid
α
α+β
0
β
γ
β+γ
XB
1
In simple eutectic systems, discussed above, there are only two
solid phases (α and β) that exist near the ends of phase
diagrams.
Phases that are separated from the composition extremes (0%
and 100%) are called intermediate phases. They can have
crystal structure different from structures of components A and
B.
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
T
liquid
α
α+β
0
β
γ
β+γ
XB
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
1
T
liquid
α
α+β
0
β
γ
β+γ
XB
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
1
ΔHmix<0 - tendency to form high-melting point intermediate phase
T
liquid
α
0
1
XB
Increasing
negative ΔHmix
liquid
T
α
α`
0
1
XB
T
liquid
α
β
α+β
0
γ
β+γ
1
MSE 3050, Phase Diagrams
XBand 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 3050, 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
intermetallic
compound
This diagram can be thought of as two joined eutectic diagrams,
for Mg-Mg2Pb and Mg2Pb-Pb. In this case compound Mg2Pb
(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 theandsystem.
MSE
3050, Phase
Diagrams
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).
T
T
liquid
liquid
α+l
l+γ
α+γ
γ
0
β
+
l
α
β
l+AB
α + AB
AB
α
α+l
β+γ
1
XB
0
non-stoichiometric
β
+
l
β + AB
1
XB
stoichiometric
Common stoichiometric compounds:
compound
AB
A6B5 A5B4 A4B3 A3B2 A5B3
composition, at%
50.0
45.5
44.4
42.9
40.0
37.5
compound
A2B A5B2
A3B
A4B
A5B
A6B
composition, at%
33.3
25.0
20.0
16.7
14.3
28.6
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
β
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.
Cu-Zn
Eutectoid
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Eutectic and Eutectoid Reactions
Temperature
l
γ
Eutectoid
temperature
α
γ +l
Eutectic
temperature
l +β
γ +β
α+γ
β
α +β
Eutectoid
composition
Eutectic
composition
Composition
The above phase diagram contains both an eutectic reaction and
its solid-state analog, an eutectoid reaction
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Temperature
l
γ
Eutectoid
temperature
α
α+γ
γ +l
l +β
γ +β
β
α +β
Composition
MSE 3050, 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
Temperature
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.
α + liquid
α +β
β
liquid
β + liquid
Peritectoid is a three-phase reaction similar to peritectic but
occurs from two solid phases to one new solid phase (α + β = γ).
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Temperature
α + liquid
α +β
β
liquid
β + liquid
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Example: The Iron–Iron Carbide (Fe–Fe3C) 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 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Phases in Fe–Fe3C Phase Diagram
¾ α-ferrite - solid solution of C in BCC Fe
• Stable form of iron at room temperature.
• The maximum solubility of C is 0.022 wt%
• Transforms to FCC γ-austenite at 912 °C
¾ γ-austenite - solid solution of C in FCC Fe
• The maximum solubility of C is 2.14 wt %.
• Transforms to BCC δ-ferrite at 1395 °C
• Is not stable below the eutectic temperature
(727 ° C) unless cooled rapidly
¾ δ-ferrite solid solution of C in BCC Fe
• The same structure as α-ferrite
• Stable only at high T, above 1394 °C
• Melts at 1538 °C
¾ Fe3C (iron carbide or cementite)
• This intermetallic compound is metastable, it remains
as a compound indefinitely at room T, but decomposes
(very slowly, within several years) into α-Fe and C
(graphite) at 650 - 700 °C
¾ Fe-C liquid solution
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
C is an interstitial impurity in Fe. It forms a solid solution with
α, γ, δ phases of iron
Maximum solubility in BCC α-ferrite is limited (max. 0.022 wt%
at 727 °C) - BCC has relatively small interstitial positions
Maximum solubility in FCC austenite is 2.14 wt% at 1147 °C FCC has larger interstitial positions
Mechanical properties: Cementite is very hard and brittle - can
strengthen steels. Mechanical properties of steel depend on the
microstructure, that is, how ferrite and cementite are mixed.
Magnetic properties: α -ferrite is magnetic below 768 °C,
austenite is non-magnetic
Classification. Three types of ferrous alloys:
• Iron: less than 0.008 wt % C in α−ferrite at room T
• Steels: 0.008 - 2.14 wt % C (usually < 1 wt % )
α-ferrite + Fe3C at room T
Examples of tool steel (machine tools for cutting other metals):
Fe + 1wt % C + 2 wt% Cr
Fe + 1 wt% C + 5 wt% W + 6 wt % Mo
Stainless steel (food processing equipment, knives,
petrochemical equipment, etc.): 12-20 wt% Cr, ~$1500/ton
• Cast iron: 2.14 - 6.7 wt % (usually < 4.5 wt %)
heavy equipment casing
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Eutectic and eutectoid reactions in Fe–Fe3C
Eutectic: 4.30 wt% C, 1147 °C
L ↔ γ + Fe3C
Eutectoid: 0.76 wt%C, 727 °C
γ(0.76 wt% C) ↔ α (0.022 wt% C) + Fe3C
Eutectic and eutectoid reactions are very important in heat
treatment of steels
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Development of Microstructure in Iron - Carbon alloys
Microstructure depends on composition (carbon content) and
heat treatment. In the discussion below we consider slow
cooling in which equilibrium is maintained.
Microstructure of eutectoid steel (I)
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Microstructure of eutectoid steel (II)
When alloy of eutectoid composition (0.76 wt % C) is cooled
slowly it forms perlite, a lamellar or layered structure of two
phases: α-ferrite and cementite (Fe3C)
The layers of alternating phases in pearlite are formed for the
same reason as layered structure of eutectic structures:
redistribution C atoms between ferrite (0.022 wt%) and cementite
(6.7 wt%) by atomic diffusion.
Mechanically, pearlite has properties intermediate to soft, ductile
ferrite and hard, brittle cementite.
In the micrograph, the dark areas are
Fe3C layers, the light phase is αferrite
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Microstructure of hypoeutectoid steel (I)
Compositions to the left of eutectoid (0.022 - 0.76 wt % C)
hypoeutectoid (less than eutectoid -Greek) alloys.
γ → α + γ → α + Fe3C
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Microstructure of hypoeutectoid steel (II)
Hypoeutectoid alloys contain proeutectoid ferrite (formed above
the eutectoid temperature) plus the eutectoid perlite that contain
eutectoid ferrite and cementite.
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Microstructure of hypereutectoid steel (I)
Compositions to the right of eutectoid (0.76 - 2.14 wt % C)
hypereutectoid (more than eutectoid -Greek) alloys.
γ → γ + Fe3C → α + Fe3C
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Microstructure of hypereutectoid steel
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Microstructure of hypereutectoid steel (II)
Hypereutectoid alloys contain proeutectoid cementite (formed
above the eutectoid temperature) plus perlite that contain
eutectoid ferrite and cementite.
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
How to calculate the relative amounts of proeutectoid phase
(α or Fe3C) and pearlite?
Application of the lever rule with tie line that extends from the
eutectoid composition (0.75 wt% C) to α – (α + Fe3C) boundary
(0.022 wt% C) for hypoeutectoid alloys and to (α + Fe3C) – Fe3C
boundary (6.7 wt% C) for hipereutectoid alloys.
Fraction of α phase is determined by application of the lever rule
across
the
entire
(α +and
Fe3Kinetics,
C) phase
field:
MSE 3050,
Phase
Diagrams
Leonid
Zhigilei
Example for hypereutectoid alloy with composition C1
Fraction of pearlite:
WP = X / (V+X) = (6.7 – C1) / (6.7 – 0.76)
Fraction of proeutectoid cementite:
WFe3C = V / (V+X) = (C1 – 0.76) / (6.7 – 0.76)
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
The most important phase diagram in the history of civilization
pearlite
naturally formed composite:
hard & brittle ceramic (Fe3C)
+
soft BCC iron (ferrite)
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Reconstruction of Baroque organs (TRUESOUND EU project)
Baroque-style organ reconstructed in
Örgryte Nya Kyrka, Gothenburg, Sweden
Components of reed pipes
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
MRS Bulletin
Vol 32, March
2007, 249-255
Reconstruction of Baroque organs (TRUESOUND EU project)
nondestructive measurements of
composition on ~30 organs
produced from 1624 to 1880
(neutron, X-ray diffraction, SIMS)
• Zn is soluble in Cu
• Pb is insoluble in Cu and is
distributed non-uniformly in
the alloy
Cu – Zn phase diagram
Cu – Pb phase diagram
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Reconstruction of Baroque organs (TRUESOUND EU project)
Surprisingly constant Zn
concentration:
• ~26 wt.% from 1624 to 1790
• ~32.5 wt.% from 1750
All historical tongues and
shallots are made of α-brass
(fcc solid solution of Zn in Cu)
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Reconstruction of Baroque organs (TRUESOUND EU project)
Before 1738 Zn was only available as zinc oxide (ZnO) or zinc carbonate
(ZnCO3) and brass was produced by mixing solid Cu and gas-phase Zn in a
process called cementation:
pieces of Cu, coal, and ZnCO3 are heated to above Tb of Zn (907ºC) but below
Tm of Cu (1083ºC)
ZnCO3 is reduced and evaporated Zn diffuses into hot solid Cu
Heating much above 907ºC was not economical – temperature was kept slightly
above this temperature – experienced craftsman can estimate temperature from
the color of a furnace within ~20ºC – the process was taking place at ~920ºC
Zn concentration in brass is determined by solidus concentration – maximum
solubility of Zn in solid Cu (26 wt.% at 920ºC)
1738: patent on distilling metallic Zn from ZnCO3, 1781: patent on alloying brass
from metallic Zn – concentration of Zn in Cu is no longer limited by the
MSE 3050,at
Phase
Diagrams
and Kinetics,
Leonid Zhigilei
solubility
920ºC,
but is 32.5
wt.% - maximum
solubility after crystallization.
The Gibbs phase rule (I)
Let’s consider a simple one-component system.
P
solid
In the areas where only one phase
is stable both pressure and
temperature can be independently
varied without upsetting the phase
equilibrium → there are 2 degrees
of freedom.
liquid
gas
T
Along the lines where two phases coexist in equilibrium, only
one variable can be independently varied without upsetting the
two-phase equilibrium (P and T are related by the Clapeyron
equation) → there is only one degree of freedom.
At the triple point, where solid liquid and vapor coexist any
change in P or T would upset the three-phase equilibrium →
there are no degrees of freedom.
In general, the number of degrees of freedom, F, in a system that
contains C components and can have Ph phases is given by the
Gibbs phase rule:
F = C − Ph + 2
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
The Gibbs phase rule (II)
Let’s now consider a multi-component system containing C
components and having Ph phases.
A thermodynamic state of each phase can be described by
pressure P, temperature T, and C - 1 composition variables. The
state of the system can be then described by Ph×(C-1+2)
variables.
But how many of them are independent?
The condition for Ph phases to be at equilibrium are:
Tα = Tβ = Tγ = …
- Ph-1 equations
Pα = Pβ = Pγ = …
- Ph-1 equations
μ αA = μ βA = μ Aγ = ...
- Ph-1 equations
μ = μ = μ = ...
- Ph-1 equations
α
B
β
B
γ
B
C sets of equations
Therefore we have (Ph – 1)×(C + 2) equations that connect the
variables in the system.
The number of degrees of freedom is the difference between the
total number of variables in the system and the minimum number
of equations among these variables that have to be satisfied in
order to maintain the equilibrium.
F = Ph(C + 1) − (Ph − 1)(C + 2) = C − Ph + 2
F = C − Ph + 2 - Gibbs phase rule
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
The Gibbs phase rule – example (an eutectic systems)
F = C − Ph + 2
P = const
F = C − Ph + 1
C=2
F = 3 − Ph
¾ In one-phase regions of the phase diagram T and XB can be
changed independently.
¾ In two-phase regions, F = 1. If the temperature is chosen
independently, the compositions of both phases are fixed.
¾ Three phases (L, α, β) are in equilibrium only at the eutectic
point in this two component system (F = 0).
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Multicomponent systems (I)
The approach used for analysis of binary systems can be
extended to multi-component systems.
wt. %
wt. %
wt. %
Representation of the composition in a ternary system (the Gibbs
triangle). The total length of the red lines is 100% :
XA +XB + XC =1
MSE 3050, 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.
G
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 3050, 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.
G
γ
α
β
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).
Eutectic point: four-phase equilibrium between α, β, γ, and liquid
MSE 3050, 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 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Another example of ternary phase diagram:
oil – water – surfactant system
Drawing by Carlos Co, University of Cincinnati
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.
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei
Summary
Make sure you understand
language and concepts:
¾
¾
¾
¾
¾
¾
¾
¾
¾
¾
¾
¾
¾
¾
¾
¾
¾
¾
¾
¾
¾
Elements of phase diagrams:
Eutectic (L → α + β)
L
α
β
α+β
Common tangent construction
Separation into 2 phases
Eutectic structure
Eutectoid (γ → α + β)
Composition of phases
γ
Weight and atom percent
α
β
Miscibility gap
α+β
Solubility dependence on T
Intermediate solid solution
Peritectic (α + L → β)
Compound
α+L
Isomorphous
L
α
Tie line, Lever rule
β
Liquidus & Solidus lines
Peritectoid (α + β → γ)
Microconstituent
α+β
Primary phase
α
β
Solvus line, Solubility limit
γ
Austenite, Cementite, Ferrite
Compound, AnBm
Pearlite
Hypereutectoid alloy
Hypoeutectoid alloy
A useful link:
Ternary alloys
http://www.soton.ac.uk/~pasr1/index.htm
Gibbs phase rule
MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei