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Binary Phase Diagram and Gibbs Free Energy - Lecture Slides | MSE 3050, Study notes of Materials science

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, peritectoidreactions)
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
B
X1
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
GP
P
P
2
2
=
=
mixmixBBAA S TΔΔHGXGXG
+
+
=
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Binary solutions with unlimited solubility
B
X1
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
¾T1is 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.
liquid
G
1
T
[
]
BBAABBAA
id lnXXlnXXRTGXGXG +++=
solid
B
G
solid
A
G
MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei
Binary solutions with unlimited solubility (II)
B
X1
0
solid
B
G
Decreasing the temperature below T1will have two effects:
will increase more rapidly than
liquid
B
G
solid
G
¾Eventually we will reach T2 melting point of pure
component A, where
liquid
G
2
T
liquid
B
liquid
AG and G solid
A
G
solid
B
G and Why?
The curvature of the G(XB) curves will decrease. Why?
solid
A
liquid
AG G =
solid
A
liquid
AG G =
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff

Partial preview of the text

Download Binary Phase Diagram and Gibbs Free Energy - Lecture Slides | MSE 3050 and more Study notes Materials science in PDF only on Docsity!

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

G A

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

G B

G^ 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:

G

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

GA

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

GB

Gsolid

¾ T 1 is above the equilibrium melting temperatures of both

pure components: T 1 > T (^) m(A) > T (^) m(B) → the liquid phase will be the stable phase for any composition.

Gliquid

T 1

G id^ =XAGA+XBGB+RT [X AlnXA+XBlnXB]

solid

GB

solid

GA

MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei

Binary solutions with unlimited solubility (II)

solid

GB

Decreasing the temperature below T 1 will have two effects:

™ will increase more rapidly than

liquid

GB

Gsolid

¾ Eventually we will reach T 2 – melting point of pure

component A, where

Gliquid

T 2

liquid B

liquid

G A andG

solid

GA

solid

and GB Why?

™ The curvature of the G(XB ) curves will decrease. Why?

solid A

liquid

G A =G

solid A

liquid

G A =G

MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei

Binary solutions with unlimited solubility (III)

solid

GB

¾ For even lower temperature T 3 < T 2 = T m(A) the Gibbs free

energy curves for the liquid and solid phases will cross.

liquid

GB

Gsolid

Gliquid

T 3

solid

GA

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

GA

MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei

Binary solutions with unlimited solubility (IV)

liquid

GA

solid B

liquid

G B =G

G^ solid

Gliquid

T 4

solid

GA

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

well as points X 1 and X 2 are shifting to the right, until, at T 4

= T (^) m(B) the curves will intersect at X 1 = X 2 = 1

liquid B

liquid

GA andG

solid B

solid

GA andG

MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei

Binary solutions with unlimited solubility (V)

solid

GB

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

liquid

GB

Gsolid

Gliquid

T 3 solid

GA

0 XB 1

solid solid + liquid liquid

liquid

GA

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 %:

C A C A

C A

C

B

wt A A

wt B

A

wt at B

B ×

C A C A

C A

C

B

wt A A

wt B

B

wt at A

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)

CA C A

C A

C

A

at B A

at B

B

at wt B B ×

CA C A

C A

C

A

at B A

at B

A

at wt A A ×

MSE 305, Phase Diagrams and Kinetics, Leonid Zhigilei

Phase compositions and amounts. An example.

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

  • 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 (cored) 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 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

G^ solid

Gliquid

T 1

0 XB^1

Gsolid

T 2 <T (^1) Gliquid

0 X B^1

Gsolid

G liquid^3

T <T

0 X B^1

T

0 XB^1

T 3

T 1

T 2

G

G G

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.

Gsolid

G liquid T 1

0 XB 1

G

Gsolid

G liquid^2

T <T

0 X B 1

G

T

0 XB^1

T 3

T 1 T 2

liquid

α 1 α 1 +α 2

Gsolid

G liquid^3

T <T

0 X B^1

G

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

• 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

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.

L → α + L → α + β

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).

T

0 XB^1

liquid

α

γ (^) β + γ

l + γ

α + l

β

  • l

T

0 XB^1

liquid

α α + AB β

AB^ β^ +^ AB

l +AB

α + l

β

  • l

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

  • 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 Fe 3 C (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 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.

γ → α + γ → α + Fe 3 C

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.

γ → γ + Fe 3 C → α + Fe 3 C

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

pearlite

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.

G

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

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).

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

L

α + β

α β

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, 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