Binary phase diagrams

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,

Binary phase diagram and Gibbs free energy

B

X

1

0

α A

G

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

B

G

α

G

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: G T

β

α

S T G P − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ T c T S T G _P P P 2 2 − = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ − = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ∂ ∂ mix mix B B A A

G

X

G

ΔH

TΔ

S

X

G

=

+

+

−

Binary solutions with unlimited solubility

B

X

1

0

liquid A

G

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 B

G

solid

G

¾ T

₁ is above the equilibrium melting temperatures of both pure components: T₁ > T_m(A) > T_m(B) → the liquid phase will be the stable phase for any composition.

liquid

G

1

T

solid B

G

solid A

G

Binary solutions with unlimited solubility (II)

B

X

1

0

solid B

G

Decreasing the temperature below T₁ will have two effects: ™ will increase more rapidly than

liquid B

G

solid

G

¾ Eventually we will reach

T

₂ – melting point of pure component A, where liquid

G

2

T

liquid B liquid A

and

G

G

G

solid_A solid B

G

and

_Why?

™ The curvature of the G(X_B) curves will decrease.

Why?

solid A liquid A

G

G

=

solid A liquid A

G

G

=

Binary solutions with unlimited solubility (III)

solid B

G

¾ For even lower temperature

T

₃ < T₂ = T_m(A) the Gibbs free

energy curves for the liquid and solid phases will cross.

liquid B

G

solid

G

liquid

G

3

T

solid A

G

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.

B

X

1

0

solid liquid solid + liquid 1

X

X

₂ liquid A

G

Binary solutions with unlimited solubility (IV)

liquid A

G

solid B liquid B

G

G

=

solid

G

liquid

G

4

T

solid A

G

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.

B

X

1

0

As temperature decreases below T₃ continue to increase more rapidly than

¾ Therefore, the intersection of the Gibbs free energy curves, as well as points X₁ and X₂ are shifting to the right, until, at

T

₄

= T_m(B) the curves will intersect at X₁ = X₂ = 1

liquid B liquid A

and

G

G

solid B solid A

and

G

G

Binary solutions with unlimited solubility (V)

solid B

G

Based on the Gibbs free energy curves we can now construct a phase diagram for a binary isomorphous systems

liquid B

G

solid

G

liquid

G

3

T

solid A

G

B

X

1

0

solid liquid solid + liquid liquid A

G

2

T

3

T

4

T

5

T

1

T

T

4

T

2

T

1

T

s

s

s

l

l

l

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

Solidus line

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.

α + L

α

L

_{liquid solution} liquid solution + crystallites of solid solution polycrystal solid solution

Binary solutions with unlimited solubility (VII)

Liquidus

Solidus

A 20 40 60 80 B

Composition, wt %

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.

B

X

solid B

X

liquid B

X

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

The lever rule is a mechanical analogy to the mass balance calculation. The tie line in the two-phase region is analogous to a lever balanced on a fulcrum.

Interpretation of Phase Diagrams: the Lever Rule

1) All material must be in one phase or the other: W_α+ W_β = 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: W_αC_α + W_βC_β = C_o

Derivation of the lever rule:

α

W

α

α

+

β

β

β

W

Composition/Concentration: weight fraction vs. molar fraction

Composition can be expressed in

Molar fraction, X_B, 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

Where n_mA _{and n}

mB 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

100

m

m

m

C

A B B wt

_×

+

=

100

n

n

n

C

_A m B m B m at

_×

+

=

where m_A and m_B are the weights of the components in the system.

100

X

C

at

=

_B

×

B B B m

A

m

n

=

where AA and AB are atomic

weights of elements A and B. A A A m

A

m

n

=

Composition Conversions Weight % to Atomic %: Atomic % to Weight %:

100

A

C

A

C

A

C

C

B wt A A wt B A wt B at B

×

+

=

100

A

C

A

C

A

C

C

B wt A A wt B B wt A at A

×

+

=

W_L = (Cwt α - Cwto) / (Cwtα - CwtL)

Of course the lever rule can be formulated for any specification

of composition:

M

L

_{= (X}

Bα

- X

B0

)/(X

Bα

- X

BL

) = (C

atα

- C

ato

) / (C

atα

- C

atL

)

M

α

= (X

_B0

_{- X}

BL

)/(X

Bα

- X

BL

) = (C

at0

- C

atL

) / (C

atα

- C

atL

)

100

A

C

A

C

A

C

C

A at A B at B B at B wt B

×

+

=

100

A

C

A

C

A

C

C

A at A B at B A at A wt A

×

+

=

Phase compositions and amounts. An example.

Mass fractions: W_L = S / (R+S) = (C_α - C_o) / (C_α- C_L) = 0.68

W_α = R / (R+S) = (C_o - C_L) / (C_α- C_L) = 0.32 C_o = 35 wt. %, C_L = 31.5 wt. %, C_α = 42.5 wt. %

Development of microstructure in isomorphous alloys

Equilibrium (very slow) cooling

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.

Development of microstructure in isomorphous alloys

Non-equilibrium cooling

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.

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 ΔH_mix > 0

solid

G

liquid

G

1

T

B

X

1

0

solid

G

liquid

G

1 2

T

T

<

B

X

1

0

solid

G

liquid

G

T

3

<

T

2 B

X

1

0

T

B

X

1

0

3

T

1

T

2

T

G

G

G

liquid α α₁+α₂

At low temperatures, there is a region where the solid solution is most stable as a mixture of two phases α₁ and α₂ with

T

B

X

1

0

1

T

2

T

liquid α α₁+α₂ B

X

1

0

G

B

X

1

0

G

[

A A B B

]

B A B B A A reg

_X

_G

_X

_G

_ΩX

_X

_RT

_X

_lnX

_X

_lnX

G

=

+

+

+

+

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. solid

G

liquid

G

T

1 B

X

1

0

G

solid

G

liquid

G

T

2

<

T

1 B

X

1

0

G

T

B

X

1

0

3

T

1

T

2

T

liquid α₁ α₁+α₂ solid

G

liquid

G

T

3

<

T

2 B

X

1

0

G

α₂ α₁+l α₂+l

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.

1

T

B

X

1

0

G

_T

₂

_<

_T

₁

T

B

X

1

0

3

T

1

T

2

T

liquid α α + β β

β

α

liquid B

X

1

0

G

β

α

liquid 2 3

T

T

<

B

X

1

0

G

β

α

liquid α+l _β+l

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.

The closer is the minimum of the Gibbs free energy curve Gα_(X

B) 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_(X B).

T

B

X

1

0

1

T

liquid α α + β β 1

T

B

X

1

0

G

β

α

liquid α+l _β+l

[

_{A A B B}

]

B A B B A A reg

_X

_G

_X

_G

_ΩX

_X

_RT

_X

_lnX

_X

_lnX

G

=

+

+

+

+

0 dX dG B = _{- Minimum of G(X} B)

Temperature dependence of solubility (II)

(

1-XB

)

GA + XBGB +Ω

(

1-XB

)

XB +RT

[

(

1-XB

) (

ln 1-XB

)

+XBlnXB

]

=

(

) (

₍

)

₎

_⎥ = ⎦ ⎤ ⎢ ⎣ ⎡ + + − + − Ω + + = B B B B B B B B A B X X lnX X -1 X -1 X -1 ln -RT X 2Ω G -G dX dG

(

)

[

+

]

= + − Ω + + =-G_A G_B 2ΩX_B RT-ln1-X_B lnX_B

Solid solubility of B in α increases exponentially with temperature

[

_{A A B B}

]

B A B B A A reg

_X

_G

_X

_G

_ΩX

_X

_RT

_X

_lnX

_X

_lnX

G

=

+

+

+

+

(

)

G -G RTln

( )

X 0 X -1 X RTln 2X 1 G -G _{B A B} B B B A B ⎟⎟≈ +Ω+ = ⎠ ⎞ ⎜⎜ ⎝ ⎛ + − Ω + = if X_B is small (X_B → 0) (G_B >> G_A) Minimum of G(X_B) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛₋ − +Ω ≈ RT G G exp X B A B

Temperature dependence of solubility (III)

⎟ ⎠ ⎞ ⎜ ⎝ ⎛₋ − +Ω ≈ RT G G exp X B A B X_B vs. T X_B T_e T T X_B supersaturation with B

→ phase separation α and β

→ precipitation of B-rich intermediate phase or compound B X

₁

0 1

T

liquid α α + β β α+l _β+l T vs. X_B

Eutectic systems -

alloys with limited solubility (I)

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

Copper – Silver phase diagram liquid α + β Solidus Liquidus Solvus Temperature, °C Composition, wt% Ag

Eutectic or invariant point - Liquid and two solid phases

co-exist in equilibrium at the eutectic composition C_E and the eutectic temperature T_E.

Eutectic isotherm - the horizontal solidus line at T_E.

Eutectic reaction – transition between liquid and mixture of two

solid phases, α + β at eutectic concentration C_E.

The melting point of the eutectic alloy is lower than that of the Lead – Tin phase diagram

Invariant or eutectic point

Eutectic isotherm

Eutectic systems -

alloys with limited solubility (II)

Temperature,

°C

Compositions and relative amounts of phases are determined from the same tie lines and lever rule, as for isomorphous alloys

• C

For points A, B, and C calculate the compositions (wt. %) and relative amounts (mass fractions) of phases present.

• B

• A

Eutectic systems -

alloys with limited solubility (III)

Composition, wt% Sn

Temperature,

• C

Composition, wt% Sn

Temperature,

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,

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

No changes above the eutectic temperature T_E. At T_E all the liquid transforms to α and β phases (eutectic reaction).

→ α

β

Development of microstructure in eutectic alloys (III)

Solidification at the eutectic composition

Composition, wt% Sn

Temperature,

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

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

Temperature,

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.

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):

W

_e

= P / (P+Q)

(eutectic)

W

_α’

= Q / (P+Q)

(primary)

Temperature,

Composition, wt% Sn

Temperature,

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:

W

_α

= (Q+R) / (P+Q+R)

(α phase)

W

_β

= P / (P+Q+R)

(β phase)

Temperature,

T

B

X

1

0

T

B

X

1

0

α α + β β β+L₂ α + L₂ L₁ + L₂ α + L₁ L₁ L₂

Monotectic system

Monotectic point Upper solidus Lower solidus Monotectic isotherm Limit of liquid immiscibility

F.N. Rhines, Phase Diagrams in Metallurgy, 1956

Lower liquidus

Binary solutions with

ΔH_mix < 0 -

ordering

If ΔH_mix < 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 → |ΔH_mix| can become larger than |ΩX_AX_B| and the Gibbs free energy curve for the solid phase can become steeper than the one for liquid.

α

G

liquid

G

1

T

B

X

1

0

2 3

T

T

<

T

B

X

1

0

3

T

1

T

2

T

G

G

liquid α α` liquid

G

B

X

1

0

1 2

T

T

<

G

G

liquid B

X

1

0

α

G

α

G

α`

G

Binary solutions with

ΔH_mix < 0 -

intermediate phases

If attraction between unlike atoms is very strong, the ordered phase may extend up to the liquid.

T

B

X

1

0

liquid α β α + β γ β + γ

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.

T

B

X

1

0

liquid α β α + β γ β + γ

T

B

X

1

0

liquid α β α + β γ β + γ

ΔH_mix<0 - tendency to form high-melting point intermediate phase

T

B

X

1

0

liquid α α`

T

liquid α _β α + β _{β + γ} γ

T

B

X

1

0

liquid α Increasing negative ΔH_mix

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.

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-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

intermetallic

compound

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

B

X

1

0

liquid α β α +

γ

γ _{β + γ} l +

γ

α +

l

β+l

T

B

X

1

0

liquid α β α + AB AB β + AB l+AB α +

l

β+l

stoichiometric

non-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

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

Eutectic and Eutectoid Reactions

The above phase diagram contains both an eutectic reaction and its solid-state analog, an eutectoid reaction

α β α+

β

l

+

γ

β

γ

+

γ α+ β + l

γ

l

Temperature Eutectic temperature Eutectoid temperature Eutectoid

composition compositionEutectic Composition

α β α+

β

l

+

γ

β

γ

+

γ α+ β + l

γ

l

Temperature Eutectoid temperature Composition

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

liquid

+

α

β

α

+

β

_β

₊

_liquid

liquid

Temperature

liquid

+

α

β

α

+

β

_β

₊

_liquid

liquid

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.

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

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 %)

Eutectic and eutectoid reactions in Fe–Fe₃C

Eutectoid: 0.76 wt%C, 727 °C

γ(0.76 wt% C) ↔ α (0.022 wt% C) + Fe

₃

C

Eutectic: 4.30 wt% C, 1147 °C

L

↔ γ + Fe

₃

C

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.

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.

Microstructure of eutectoid steel (II)

In the micrograph, the dark areas are Fe₃C layers, the light phase is α-ferrite

Compositions to the left of eutectoid (0.022 - 0.76 wt % C)

hypoeutectoid (less than eutectoid -Greek) alloys.

γ → α + γ → α + Fe

₃

C

Hypoeutectoid alloys contain proeutectoid ferrite (formed above the eutectoid temperature) plus the eutectoid perlite that contain eutectoid ferrite and cementite.

Compositions to the right of eutectoid (0.76 - 2.14 wt % C)

hypereutectoid (more than eutectoid -Greek) alloys.

γ → γ + Fe

₃

C

→ α + Fe

₃

C

Hypereutectoid alloys contain proeutectoid cementite (formed above the eutectoid temperature) plus perlite that contain eutectoid ferrite and cementite.

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.

Example for hypereutectoid alloy with composition C₁

Fraction of pearlite:

W_P = X / (V+X) = (6.7 – C₁) / (6.7 – 0.76)

Fraction of proeutectoid cementite:

The most important phase diagram in the history of civilization

pearlite

naturally formed composite: hard & brittle ceramic (Fe₃C)

+

Reconstruction of Baroque organs

(TRUESOUND EU project)

Baroque-style organ reconstructed in Örgryte Nya Kyrka, Gothenburg, Sweden

nondestructive measurements of composition on ~30 organs produced from 1624 to 1880 (neutron, X-ray diffraction, SIMS)

Reconstruction of Baroque organs

(TRUESOUND EU project)

• 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

Surprisingly constant Zn concentration:

Reconstruction of Baroque organs

(TRUESOUND EU project)

• ~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)

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 T_b of Zn (907ºC) but below T_mof 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)

Let’s consider a simple one-component system.

The Gibbs phase rule (I)

liquid solid

gas

T P

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.

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:

2

Ph

C

Let’s now consider a multi-component system containing C components and having Ph phases.

The Gibbs phase rule (II)

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: T_α = T_β = T_γ = … - Ph-1 equations P_α = P_β = P_γ = … - Ph-1 equations

2

Ph

C

2)

1)(C

(Ph

1)

Ph(C

F

=

+

−

−

+

=

−

+

...

μ

μ

μ

γ A β A α A

=

=

=

- Ph-1 equations

...

μ

μ

μ

γ B β B α B

=

=

=

- Ph-1 equations 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.

2

Ph

C

¾ In one-phase regions of the phase diagram T and X_B 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).

The Gibbs phase rule – example (an eutectic systems)

2

Ph

C

F

=

−

+

1

Ph

C

F

=

−

+

const

P

=

Ph

3

F

=

−

2

C

=

Multicomponent systems (I)

The approach used for analysis of binary systems can be extended to multi-component systems.

Representation of the composition in a ternary system (the Gibbs triangle). The total length of the red lines is 100% :

X_A +X_B + X_C =1 wt. %

wt. % wt. %

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

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

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).

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).

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.

Summary

Elements of phase diagrams:

α + β β α L α + β β α γ α + L α β L Eutectic (L → α + β) Eutectoid (γ → α + β) Peritectic (α + L → β) α + β α γ β Peritectoid (α + β → γ) Compound, A_nB_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