1

CHAPTER 8

Phase Diagrams

Week Topic Learning Outcomes 5 Phase Diagrams

•  Definition and basic concepts •  Gibbs Phase Rules •  Phase diagram of pure

substance •  Binary phase diagram – binary

isomorphous and binary eutectic

•  Lever Rule

(Sections 8.1 – 8.7, omit 8.3 and 8.6)

It is expected that students are able to:

•  describe phase diagram of a material system and apply Gibbs phase rule

•  describe binary isomorphous and binary eutectic phase diagrams

•  draw generic diagram showing all phase regions and relevant information

•  determine phase composition and phase fraction in a mixture using tie-line and lever rule

•  describe microstructure evolution during equilibrium cooling as metal solidifies

Learning Outcome

Phase diagram is a graphical representation of what phases are present in a material system at various temperature, pressure and composition.

Phase diagram for a mixture is plotted as temperature vs. composition. For pure substance, phase diagram is plotted as temperature versus pressure.

Phase Diagram for Alloys or Mixtures

Introduction: Phase Diagram and Terminology

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display

Introduction (cont.)

•  Information from phase diagram: -  Represents phases present in metal at different conditions

(temperature, pressure and composition). -  Indicates equilibrium solid solubility of one element in another. -  Indicates temperature range under which solidification occurs. -  Indicates temperature at which different phases start to melt.

•  Phase: a region in a material system that differs in its structure and/or composition from another region. For example, liquid water and ice are two separate phases although they both are composed of H2O. Phases are separated from each other by a phase boundary.

•  Component: element, compound or solution in the system. For example, the components of syrup are water (H2O) and sugar. The components of brass are copper and zinc. A binary alloy has two components, a ternary alloy – three, etc.

4

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display

Phase Diagram of Pure Substances

•  Pure systems (e.g. water) can exist as solid, liquid, or vapor phases, depending on temperature and pressure.

•  Pure water: Along freezing line, two phases (solid-liquid) coexist. Along vaporization line, liquid and vapor coexist.

•  At triple point, three different phases coexist.

5

8‐6

Gibb’s phase rule is an equation that gives the number of phases that can coexist in equilibrium in a system:

Degree of freedom indicates number of variables (temperature, pressure, composition) that can be changed without changing number of phases.

P = number of phases that coexist within a system C = number of components (elements, compounds) in the system F = degree of freedom for the system 2 = two independent variables (temperature, pressure)

For solid-liquid systems, pressure is assumed to be constant,

P + F = C + 1

P + F = C + 2

Gibbs Phase Rule

P + F = C + 2

A: One-phase region

P + F = C + 2 1 + F = 1 + 2; thus F = 2 Can change two variables (T and P) and still have this system in one phase.

Pure H2O: only 1 component (water) in the system. C = 1

B: Two-phase region P + F = C + 2 2 + F = 1 + 2; thus F = 1 Can change only one variable to keep system in two-phase. If a particular pressure is specified, there is only one temperature at which 2 phases can coexist.

C: Three-phase region

P + F = C + 2 3 + F = 1 + 2; thus F = 0 Invariant point. T and P cannot be changed if you want to keep this system in 3 phases.

Gibbs Phase Rule – Pure Component

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display

Binary Isomorphous Alloy System – Generic Diagram

•  Isomorphous system: Two elements completely soluble in each other in both the liquid and solid states. Ex: Cu-Ni system.

8

Three distinct phases: liquid (L); solid solution (α); and solid + liquid (α + L)

Above liquidus line, alloy exists as a liquid. We see one liquid phase.

Below solidus line, alloy is a stable solid. We see only one solid phase; one crystal structure. Cannot tell between A and B.

In 2-phase region, solid and liquid in equilibrium with one other. Composition of liquid and solid phases at any temperature can be determined by drawing a tie-line.

Tie-line connects phases in equilibrium with each other - essentially an isotherm.

Tem

pera

ture

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display

Binary Isomorphous Alloy: Cu-Ni System

9

Cu and Ni have complete liquid solubility and complete solid solubility. Complete solid solubility occurs because both have FCC crystal structures, similar atomic radii, similar EN, and same valence state. Satisfies Hume-Rothery solid solubility rules (Chap. 4)

10

•  Solubility – the ability of one element to dissolve another element.

•  Solubility limit is the maximum concentration for which only a single-phase solution occurs. Example: alcohol has unlimited solubility in water; sugar has limited solubility in water; oil is insoluble in water.

•  Same concept applies to solid phases: Cu and Ni are totally soluble in all proportions (just like water-methanol). C has a limited solubility in Fe, up to only 2.08%.

Solubility and Solubility Limit

11

Solubility Limit in Phase Diagram

• Solubility Limit: Max concentration for which

only a solution occurs.

• Ex: Water-Sugar Q: What is solubility limit at 20oC? A: 65wt% sugar. If Co < 65wt% sugar: syrup

If Co > 65wt% sugar: syrup + sugar.

• Solubility limit increases with T : e.g., if T = 100°C, solubility limit = 80wt% sugar.

Sugar/Water Phase Diagram

Suga

r

Tem

pera

ture

(°C

)

0 20 40 60 80 100 C = Composition (wt% sugar)

L (liquid solution

i.e., syrup)

Solubility Limit L

(liquid) + S

(solid sugar) 20

4 0

6 0

8 0

10 0

Wat

er

Foundations of Materials Science and Engineering, 5th Edn. Smith and Hashemi

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display

The Lever Rule

•  The lever rule gives the weight % of the phases in any two-phase regions.

Wt. fraction of solid phase:

Wt. fraction of liquid phase:

12

€

XS = w0 − wL

wS − wL

€

XL = wS − w0

wS − wL

For a given temperature and composition we can use phase diagram to determine:

a) What phases are present

b) Compositions of the phases in 2-phase region

c) Amount or weight fractions of these phases

b) Finding the composition in a two-phase region: 1.  Locate composition and temperature in diagram

2.  Draw the tie line from liquidus line to solidus line and then drop vertical lines to the composition x-axis.

3.  Composition of liquid and solid phases are given by the intersection of the vertical lines with the composition axis.

Interpretation of Phase Diagram

c) Finding the amount of phases in a two-phase region:

1.  Locate composition and temperature in diagram

2.  In two-phase region, draw the tie line

3.  Fraction of a phase is determined by taking the length of the tie line to the phase boundary for the other phase, divided by total length of the tie line.

The Lever Rule

ML Mα

R S

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

Interpretation of Phase Diagram (cont.)

Phase Diagram: # and the phases present

15 wt% Ni

20 40 60 80 100 0 1000

1100

1200

1300

1400

1500

1600 T(°C)

L (liquid)

α (FCC solid

solution)

Cu-Ni phase diagram

• If we know T and Co, then we know: -- the number and the phases present.

• Example: A (1100°C, 60): 1 phase: α

B (1250°C, 35): 2 phases: L + α

B (

1250

°C,3

5)

A(1100°C,60)

16

wt% Ni

20

1200

1300

T(°C)

L (liquid)

α (solid)

30 40 50

Cu-Ni system

• If we know T and Co, then we know: -- the composition of each phase.

• Example: T A A

35 C o

32 C L

At T A = 1320°C: Only Liquid (L) C L = C o ( = 35 wt% Ni)

At T B = 1250°C: Both α and L

C L = C liquidus ( = 32 wt% Ni here) C α = C solidus ( = 43 wt% Ni here)

At T D = 1190°C: Only Solid ( α ) C α = C o ( = 35 wt% Ni )

C o = 35 wt% Ni

B T B

D T D

tie line

4 C α

3

Phase Diagram: Compositions of the phases

17

• If we know T and Co, then we know: -- the amount of each phase (given in wt%).

• Example:

At T A : Only Liquid (L) W L = 100 wt%, W α = 0

At T D : Only Solid ( α ) W L = 0, W α = 100 wt%

C o = 35 wt% Ni

wt% Ni

20

1200

1300

T(°C)

L (liquid)

α (solid)

3 0 4 0 5 0

Cu-Ni system

T A A

35 Co 32

CL

B T B

D T D

tie line

43 Cα

R S At T B : Both α and L

€

=43−3543−32

= 73 wt%

= 27 wt%

WL = S R + S

Wα = R R + S

Phase Diagram: Amount of the phases

•  In pure component system, melting occurs at a well-defined melting temperature (at mp). Consider the temperature y-axis.

•  In multi-component (alloy) systems, melting occurs through a range of temperatures, between the solidus and liquidus lines. Solid and liquid phases are in equilibrium in this temperature range.

Binary Isomorphous Alloy: Melting and Cooling

19

•  Solidification in the 2-phase region (liquid + solid) occurs gradually upon cooling from liquidus line.

•  Composition of the solid and the liquid change gradually during cooling (determined by the tie-line method).

•  Nuclei of the solid phase form and they grow to consume all the liquid at the solidus line

Microstructure Development in Binary Isomorphous Alloy During Slow (Equilibrium) Cooling

20

wt% Ni 20

120 0

130 0

3 0 4 0 5 0 110 0

L (liquid)

α (solid)

T(°C)

A

35 C o

L: 35wt%Ni Cu-Ni

system

• Phase diagram: Cu-Ni system.

Adapted from Fig. 9.4, Callister 7e.

• Consider Co = 35 wt%Ni.

46 35 43 32

α : 43 wt% Ni L: 32 wt% Ni

L: 24 wt% Ni α : 35 wt% Ni

B α: 46 wt% Ni L: 35 wt% Ni

C

D

E

24 35

α: 35 wt% Ni

• System is: -- binary: 2 components: Cu and Ni. -- isomorphous i.e., complete solubility of one component in another; α phase field extends from 0 to 100 wt% Ni.

Microstructure Development in Binary Isomorphous Alloy During Slow (Equilibrium) Cooling